2021 ABCD Workshop: SEM Practical
Ethan McCormick and John Flournoy
2021-07-13
1 Intro
1.1 Loading Relevant Packages
packages = c('lavaan','nlme','ggplot2','patchwork',
'kableExtra', 'psych', 'dplyr', 'tidyr',
'semPlot')
if (length(setdiff(packages, rownames(installed.packages()))) > 0) {
install.packages(setdiff(packages, rownames(installed.packages())),
repos = "http://cran.us.r-project.org")
}
invisible(lapply(packages, library, character.only = TRUE))1.2 Read In Data
ABCD = read.csv('abcd_sem.csv')
str(ABCD)## 'data.frame': 964 obs. of 13 variables:
## $ id : int 1 2 3 4 5 6 7 8 9 10 ...
## $ female: int 1 1 0 0 0 0 0 0 1 1 ...
## $ advers: int 0 0 0 1 0 1 0 1 0 1 ...
## $ VS.1 : num 6.67 5.24 6.19 6.19 7.62 ...
## $ VS.2 : num NA 4.29 9.05 4.76 5.71 ...
## $ VS.3 : num NA NA 8.57 4.76 7.14 ...
## $ VS.4 : num NA NA 7.14 6.19 8.57 ...
## $ VS.5 : num NA NA 8.82 5.31 7.08 ...
## $ EXT.1 : num 1.08 3.25 2.17 NA 3.25 ...
## $ EXT.2 : num 0 2.17 2.17 2.53 2.53 ...
## $ EXT.3 : num NA 1.083 2.528 0.722 3.25 ...
## $ EXT.4 : num NA NA 2.89 1.08 3.25 ...
## $ EXT.5 : num NA NA 4.494 0.809 1.606 ...These data have been synthesized from other data and then addition simulations were performed to make the data behave. The original data labels were changed for use in the workshop. The relevant variables include:
id: unique identifier
female: self-identified sex (binary: 0 = male, 1 = female)
advers: did the individual experience early-childhood adversity (binary: 0 = no, 1 = yes)
VS: measures of ventral striatum response during reward anticipation
EXT: measures of parent-reported externalizing behavior
1.3 Descriptives
describe(ABCD[,2:ncol(ABCD)], fast = TRUE) %>%
kbl() %>%
kable_styling(full_width = F)| vars | n | mean | sd | min | max | range | se | |
|---|---|---|---|---|---|---|---|---|
| female | 1 | 964 | 0.4460581 | 0.4973398 | 0.000000 | 1.000000 | 1.000000 | 0.0160182 |
| advers | 2 | 964 | 0.5477178 | 0.4979762 | 0.000000 | 1.000000 | 1.000000 | 0.0160387 |
| VS.1 | 3 | 964 | 6.2393796 | 1.4674234 | 1.428571 | 10.000000 | 8.571429 | 0.0472625 |
| VS.2 | 4 | 917 | 6.4355819 | 1.5863089 | 1.904762 | 10.000000 | 8.095238 | 0.0523845 |
| VS.3 | 5 | 912 | 6.5544069 | 1.5376050 | 1.904762 | 10.000000 | 8.095238 | 0.0509152 |
| VS.4 | 6 | 880 | 6.7408009 | 1.5402031 | 1.428571 | 10.000000 | 8.571429 | 0.0519202 |
| VS.5 | 7 | 880 | 7.0536888 | 1.5951754 | 1.218616 | 12.257785 | 11.039169 | 0.0537734 |
| EXT.1 | 8 | 923 | 2.3431141 | 0.8614213 | 0.000000 | 3.972222 | 3.972222 | 0.0283540 |
| EXT.2 | 9 | 908 | 2.4371021 | 0.8133013 | 0.000000 | 3.972222 | 3.972222 | 0.0269903 |
| EXT.3 | 10 | 884 | 2.4415848 | 0.8586684 | 0.000000 | 3.972222 | 3.972222 | 0.0288801 |
| EXT.4 | 11 | 828 | 2.6071522 | 0.8143713 | 0.000000 | 3.972222 | 3.972222 | 0.0283014 |
| EXT.5 | 12 | 828 | 2.7031994 | 1.2521188 | -1.013396 | 6.267124 | 7.280519 | 0.0435141 |
#knitr::kable(describe(ABCD[,2:ncol(ABCD)], fast = TRUE))
# Density by Wave: VS Activation
ggplot(ABCD %>%
pivot_longer(cols=starts_with('VS'),
names_to='wave',
values_to='VS'),
aes(x=VS, group=wave, fill=wave)) +
geom_density(alpha=.4) +
labs(title = 'VS Activation by Wave',
x='Beta',
y = 'Density',
fill='Wave:') +
scale_fill_discrete(labels = c('1','2','3','4','5')) +
theme(legend.position='bottom')
# VS Activation Change over Time: By ID
ggplot(ABCD %>%
pivot_longer(cols=starts_with('VS'),
names_to='wave',
values_to='VS') %>%
filter(id %in% sample(unique(ABCD$id), 100)),
aes(x=wave, y=VS, group=id)) +
geom_line() +
labs(title = 'Changes in VS Activation',
x='Wave',
y = 'Beta') +
scale_x_discrete(labels = c('1','2','3','4','5'))
# Density by Wave: Externalizing Behavior
ggplot(ABCD %>%
pivot_longer(cols=starts_with('EXT'),
names_to='wave',
values_to='EXT'),
aes(x=EXT, group=wave, fill=wave)) +
geom_density(alpha=.4) +
labs(title = 'Externalizing Behavior by Wave',
x='Beta',
y = 'Density',
fill='Wave:') +
scale_fill_discrete(labels = c('1','2','3','4','5')) +
theme(legend.position='bottom')
# Externalizing Behavior Change over Time: By ID
ggplot(ABCD %>%
pivot_longer(cols=starts_with('EXT'),
names_to='wave',
values_to='EXT') %>%
filter(id %in% sample(unique(ABCD$id), 100)),
aes(x=wave, y=EXT, group=id)) +
geom_line() +
labs(title = 'Changes in Externalizing',
x='Wave',
y = 'Externalizing Behavior') +
scale_x_discrete(labels = c('1','2','3','4','5'))
2 Some basics
2.1 Residualized change v difference scores
To easily compute these scores, I’ll make the data long and compute the lag (t-1) value for each variable.
ABCD_l <- tidyr::pivot_longer(ABCD, names_to = 'key', values_to = 'value', tidyr::matches('^(VS|EXT)')) %>%
tidyr::extract(col = key, into = c('var', 'wave'), regex = '(\\w+)\\.(\\d+)') %>%
tidyr::pivot_wider(names_from = 'var', values_from = 'value') %>%
dplyr::group_by(id) %>%
dplyr::mutate(across(c(wave, EXT, VS), lag, .names = '{.col}_lag'))Now I’ll get the residuals from a simple lm model, and also compute the raw difference score.
#residualized change
#get the residual for the regression for each wave on the one before it
ABCD_l_resid <- dplyr::group_by(filter(ABCD_l, !is.na(wave_lag)), wave, wave_lag) %>%
mutate(VS_resid = resid(lm(VS ~ 1 + VS_lag, na.action = 'na.exclude')),
EXT_resid = resid(lm(EXT ~ 1 + EXT_lag, na.action = 'na.exclude')))
#difference scores
#just take the difference between the measure at wave T and wave T-1
ABCD_l_resid_diff <- dplyr::group_by(ABCD_l_resid, wave, wave_lag, id) %>%
mutate(VS_diff = VS - VS_lag,
EXT_diff = EXT - EXT_lag)What’s the correlation of these?
select(ungroup(ABCD_l_resid_diff), id, wave, matches('(VS|EXT)_(resid|diff)')) %>%
pivot_longer(cols = c(-id, -wave)) %>%
extract(name, c('var', 'stat'), '(\\w+)_(\\w+)') %>%
pivot_wider(names_from = 'stat', values_from = 'value') %>%
ggplot(aes(x = diff, y = resid)) +
# geom_point(size = .5, alpha = .05) +
geom_bin2d() +
geom_abline(intercept = 0, slope = 1) +
geom_line(stat = 'smooth', color = '#0099ff', size = 1) +
facet_grid(~var) +
scale_fill_gradient(low = '#660000', high = '#FFFF00') +
theme_minimal() +
coord_cartesian(x = c(-5, 5), y = c(-5, 5))## Warning: Removed 843 rows containing non-finite values (stat_bin2d).## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'## Warning: Removed 843 rows containing non-finite values (stat_smooth).
What does it seem like is going on here?
Think about how the variance differs for each type of estimate.
2.2 Equations, path diagrams, and … SYNTAX
Equations are the real deal (as real as it gets at this level of abstraction). Path diagrams and syntax are both ways of conveying the equations.
Let’s start with the first diagram we saw:
A simple SEM diagram with two X variables pointing at a Y variable
The lavaan package in R has syntax for all of the relationships here (plus more). If you recall, these are almost all just regressions. If you’re already familiar with regression syntax in R, this will look familiar to you. One big difference from other common modeling packages is that the model is expressed as a string of text. We also need to learn the opperator for “covariance” to get the path between \(x_{1i}\) and \(x_{2i}\). We also need an operator to specify the residual variance. Yay: the variance and covariance operators are the same. The intercept also gets it’s own notation (it’s not part of the regression for reasons).
- Regression:
VAR1 ~ VAR2 [+ VAR3 ...] - (Co)variance:
VAR1 ~~ VAR2 [+ VAR3 ...] - Mean/Intercept:
VAR1 ~ 1
Note: Let’s acknowledge that there is some weird stuff going on here. In this diagram y is regressed on the residual, but we only care about the residual’s variance (the regression coefficient is actually set to 1). So you may see diagrams with arrows like this, or just with double headed arrows pointing to the same variable. To go down the rabbit hole a little further, you specify variances the same way whether they are variances of residuals (for variables that are on the DV side of a regression) or whether they are variances of a variable itself (if it’s an IV). Moreover, sometimes they’re not even written into the diagram! The diagram above assumes that both \(x_{1i}\) and \(x_{2i}\) have variances as well as covary with each other.
Also Note: Intercepts refer to means when they are part of a regression (conditional means), and means refer to means when they are unconditional. The syntax for both is the same
Also Also Note: Variance/covariance structures are often very consistent across models so lavaan adds many in by default. Same with intercepts/means. For now, I’m going to write out the full model, noting where lavaan usually has defaults.
simple_sem_model <- '
y ~ x1 + x2
#lavaan covariance defaults
#notice we do not allow covariance between
#residuals and the other variables.
x1 ~~ x1
x2 ~~ x1
x2 ~~ x2
y ~~ y
#lavaan intercept defaults
#Intercept
y ~ 1
#Means
x1 ~ 1
x2 ~ 1
'We can count the number of parameters we think are implied by the model diagram and then count them in the syntax.
Let’s actually fit this model using variables from the sample data.
names(ABCD)## [1] "id" "female" "advers" "VS.1" "VS.2" "VS.3" "VS.4" "VS.5"
## [9] "EXT.1" "EXT.2" "EXT.3" "EXT.4" "EXT.5"simple_sem_model <- '
VS.2 ~ EXT.1 + EXT.2
EXT.2 ~~ EXT.1
#lavaan covariance defaults
#notice we do not allow covariance between
#residuals and the other variables.
EXT.1 ~~ EXT.1
EXT.2 ~~ EXT.1
VS.2 ~~ VS.2
#lavaan mean/intercept defaults
#Intercept
VS.2 ~ 1
#Means
EXT.1 ~ 1
EXT.2 ~ 1
'
simple_sem_fit <- lavaan::sem(simple_sem_model, data = ABCD)## Warning in lav_partable_flat(FLAT, blocks = "group", meanstructure = meanstructure, : duplicated elements in model syntax have been ignored:
## EXT.1 ~~ EXT.2summary(simple_sem_fit)## lavaan 0.6-7 ended normally after 26 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 9
##
## Used Total
## Number of observations 858 964
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.2 ~
## EXT.1 -0.155 0.079 -1.946 0.052
## EXT.2 0.435 0.082 5.336 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EXT.1 ~~
## EXT.2 0.393 0.026 14.982 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.2 5.739 0.184 31.237 0.000
## EXT.1 2.394 0.028 85.206 0.000
## EXT.2 2.459 0.027 89.839 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## EXT.1 0.678 0.033 20.712 0.000
## .VS.2 2.369 0.114 20.712 0.000
## EXT.2 0.643 0.031 20.712 0.000Let’s confirm what lavaan sets by default (see ?sem “Details” for more on the defaults).
more_simple_sem_model <- '
VS.2 ~ EXT.1 + EXT.2
EXT.2 ~~ EXT.1
'
more_simple_sem_fit <- lavaan::sem(more_simple_sem_model, data = ABCD)
summary(more_simple_sem_fit)## lavaan 0.6-7 ended normally after 23 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 6
##
## Used Total
## Number of observations 858 964
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.2 ~
## EXT.1 -0.155 0.079 -1.946 0.052
## EXT.2 0.435 0.082 5.336 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EXT.1 ~~
## EXT.2 0.393 0.026 14.982 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.2 2.369 0.114 20.712 0.000
## EXT.1 0.678 0.033 20.712 0.000
## EXT.2 0.643 0.031 20.712 0.000Wait, what about the means?
Often, we don’t even care about the means. Who interprets the intercept of a regression? The sem function is set by default to not estimate any means. But we can turn it on with meanstructure = TRUE in which case it will estimate all means and interecpts.
more_simple_sem_fit <- lavaan::sem(more_simple_sem_model, data = ABCD, meanstructure = TRUE)
summary(more_simple_sem_fit)## lavaan 0.6-7 ended normally after 26 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 9
##
## Used Total
## Number of observations 858 964
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.2 ~
## EXT.1 -0.155 0.079 -1.946 0.052
## EXT.2 0.435 0.082 5.336 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EXT.1 ~~
## EXT.2 0.393 0.026 14.982 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.2 5.739 0.184 31.237 0.000
## EXT.1 2.394 0.028 85.206 0.000
## EXT.2 2.459 0.027 89.839 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.2 2.369 0.114 20.712 0.000
## EXT.1 0.678 0.033 20.712 0.000
## EXT.2 0.643 0.031 20.712 0.000This is the syntax we’ll use to build up all the other models.
Oh, by the way, we can plot these too. More complex models don’t work well but for simple ones we can:
semPlot::semPaths(more_simple_sem_fit)
The triangle with the 1 inside indicates that a mean or intercept is estimated (this will be important for growth models where we care about the mean of the latent slope).
Crazy take-away point: You can estimate the covariance structure separate from the mean structure!
2.3 Syntax for a slightly more complicated model
A more complicated SEM diagram with two X variables pointing at two Y variables
How to think through writing out these models based on diagrams? I like to start with the regressions.
We have 2 regressions here for \(y_{1i}\) and \(y_{2i}\). Each has two arrows going into it. Let’s find where they come from and write them out in lavaan syntax:
y1 ~ x1 + x2y2 ~ y1 + x1
We now are left with one covariance:
x1 ~~ x2
Using our variables from before (even though they might not make sense), we just add one regression equation:
slightly_complicated_sem_model <- '
VS.2 ~ EXT.1 + EXT.2
VS.3 ~ VS.2 + EXT.1
EXT.2 ~~ EXT.1
'
#leaving the mean structure out
slightly_complicated_sem_fit <- sem(slightly_complicated_sem_model, data = ABCD)
summary(slightly_complicated_sem_fit)## lavaan 0.6-7 ended normally after 30 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 9
##
## Used Total
## Number of observations 820 964
##
## Model Test User Model:
##
## Test statistic 0.096
## Degrees of freedom 1
## P-value (Chi-square) 0.757
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.2 ~
## EXT.1 -0.163 0.079 -2.074 0.038
## EXT.2 0.478 0.083 5.773 0.000
## VS.3 ~
## VS.2 0.513 0.029 17.505 0.000
## EXT.1 0.080 0.054 1.470 0.141
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## EXT.1 ~~
## EXT.2 0.389 0.027 14.679 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.2 2.246 0.111 20.248 0.000
## .VS.3 1.647 0.081 20.248 0.000
## EXT.1 0.685 0.034 20.248 0.000
## EXT.2 0.621 0.031 20.248 0.000semPlot::semPaths(slightly_complicated_sem_fit, layout = 'spring')
2.4 Factors
This is where the magic starts to happen (though the above framework is awesome for doing regression too; very useful for mediation and other complex models).
Latent variables are nothing more than something invisible causing a bunch of things. In other words, it’s a bunch of different y variables being regressed on something invisible.
some_factor_model <- '
y1 ~ ETA
y2 ~ ETA
y3 ~ ETA
y4 ~ ETA
'some_factor_model <- '
EXT.1 ~ ETA
EXT.2 ~ ETA
EXT.3 ~ ETA
EXT.4 ~ ETA
'
some_factor_fit <- sem(some_factor_model, data = ABCD)
summary(some_factor_fit)This doesn’t run because ETA doesn’t exist in the data. We need to use lavaan’s syntax reserved for invisible variables:
- Factors:
FACTOR =~ VAR1 [+ VAR2 ...]
some_factor_model <- '
EXT_FAC =~ EXT.1 + EXT.2 + EXT.3 + EXT.4
'
some_factor_fit <- sem(some_factor_model, data = ABCD)
summary(some_factor_fit)## lavaan 0.6-7 ended normally after 20 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 8
##
## Used Total
## Number of observations 761 964
##
## Model Test User Model:
##
## Test statistic 18.947
## Degrees of freedom 2
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## EXT_FAC =~
## EXT.1 1.000
## EXT.2 1.001 0.053 19.055 0.000
## EXT.3 1.009 0.057 17.659 0.000
## EXT.4 1.127 0.057 19.704 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .EXT.1 0.309 0.020 15.468 0.000
## .EXT.2 0.228 0.016 13.933 0.000
## .EXT.3 0.344 0.022 15.825 0.000
## .EXT.4 0.220 0.018 12.206 0.000
## EXT_FAC 0.342 0.032 10.745 0.000Now EXT_FAC is a variable like any other, except that we’re inferring it from the observed variables (i.e., the indicators). Notice that EXT_FAC gets its own variance. We can include it in regressions, and we can also get its mean.
some_factor_model <- '
EXT_FAC =~ EXT.1 + EXT.2 + EXT.3 + EXT.4
VS.1 ~ EXT_FAC
'
some_factor_fit <- sem(some_factor_model, data = ABCD, meanstructure = TRUE)
summary(some_factor_fit)## lavaan 0.6-7 ended normally after 29 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 15
##
## Used Total
## Number of observations 761 964
##
## Model Test User Model:
##
## Test statistic 24.119
## Degrees of freedom 5
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## EXT_FAC =~
## EXT.1 1.000
## EXT.2 1.003 0.053 19.042 0.000
## EXT.3 1.009 0.057 17.617 0.000
## EXT.4 1.132 0.057 19.721 0.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.1 ~
## EXT_FAC 0.312 0.093 3.366 0.001
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .EXT.1 2.448 0.029 83.680 0.000
## .EXT.2 2.515 0.027 91.844 0.000
## .EXT.3 2.496 0.030 82.740 0.000
## .EXT.4 2.645 0.029 90.282 0.000
## .VS.1 6.227 0.050 125.176 0.000
## EXT_FAC 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .EXT.1 0.311 0.020 15.524 0.000
## .EXT.2 0.228 0.016 13.960 0.000
## .EXT.3 0.346 0.022 15.872 0.000
## .EXT.4 0.218 0.018 12.141 0.000
## .VS.1 1.850 0.095 19.444 0.000
## EXT_FAC 0.340 0.032 10.722 0.000Is it weird that the mean of EXT_FAC is 0? It’s actually the default in this kind of model to set it to 0. We’ll see later when we run a latent growth model that it’s allowed to be different from 0. However, this requires us to set other constraints.
2.4.1 “True score”
Using latent variables means we get more precision
ABCD_means <- ABCD %>%
group_by(id) %>%
mutate(EXT_MEAN = mean(c(EXT.1, EXT.2, EXT.3, EXT.4)),
VS_MEAN = mean(c(VS.1, VS.2, VS.3, VS.4)))
factor_model <- '
EXT_FAC =~ EXT.1 + EXT.2 + EXT.3 + EXT.4
VS_FAC =~ VS.1 + VS.2 + VS.3 +VS.4
VS_FAC ~~ EXT_FAC
'
factor_fit <- sem(factor_model, data = ABCD)
summary(factor_fit, stan = TRUE)## lavaan 0.6-7 ended normally after 32 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 17
##
## Used Total
## Number of observations 732 964
##
## Model Test User Model:
##
## Test statistic 78.610
## Degrees of freedom 19
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## EXT_FAC =~
## EXT.1 1.000 0.579 0.719
## EXT.2 1.006 0.054 18.677 0.000 0.583 0.780
## EXT.3 1.028 0.059 17.378 0.000 0.596 0.716
## EXT.4 1.135 0.059 19.171 0.000 0.657 0.811
## VS_FAC =~
## VS.1 1.000 0.831 0.609
## VS.2 1.202 0.092 13.068 0.000 0.999 0.669
## VS.3 1.334 0.097 13.737 0.000 1.109 0.753
## VS.4 1.200 0.092 13.002 0.000 0.998 0.663
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## EXT_FAC ~~
## VS_FAC 0.110 0.023 4.695 0.000 0.228 0.228
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .EXT.1 0.313 0.020 15.333 0.000 0.313 0.483
## .EXT.2 0.219 0.016 13.537 0.000 0.219 0.392
## .EXT.3 0.338 0.022 15.410 0.000 0.338 0.488
## .EXT.4 0.225 0.018 12.242 0.000 0.225 0.343
## .VS.1 1.173 0.074 15.821 0.000 1.173 0.629
## .VS.2 1.229 0.085 14.493 0.000 1.229 0.552
## .VS.3 0.936 0.080 11.655 0.000 0.936 0.432
## .VS.4 1.266 0.086 14.646 0.000 1.266 0.560
## EXT_FAC 0.336 0.032 10.448 0.000 1.000 1.000
## VS_FAC 0.691 0.086 7.993 0.000 1.000 1.000(scale_score_cor <- cor(ABCD_means[, c('EXT_MEAN', 'VS_MEAN')], use = 'pairwise.complete.obs'))## EXT_MEAN VS_MEAN
## EXT_MEAN 1.0000000 0.1819191
## VS_MEAN 0.1819191 1.0000000std <- standardizedSolution(factor_fit)The correlation for the latent variable model is 0.23 versus 0.18 for the zero-order correlation of the computed scale scores.
2.5 Important lavaan & semPlot stuff
2.5.1 Parameter specificaiton
We can “fix” parameters (e.g., factor loadings, covariances) by multiplying the variable by the value. To set a loading or covariance equal to 0, we can write factor =~ 0*y1 + ... or EXT ~~ 0*VS. Anything that is not freely estimated in your model or by lavaan defaults is implicitly fixed to 0. When reviewing your models you should make sure you’re okay with both the paths that do exist as well as with the paths that are left unspeficied. Sometimes fixing a path to 0 is a stronger theoretical statement than letting it be estimated freely.
Additionally you can add in equality constraints by multiplying by some label (e.g., y ~ some_label*x1, where some_label is an arbitrary string) and re-using the same label on some other parameter: y ~ some_label*x1 + some_label*x2; here both regression coefficients would be estimated exactly equal. This is a great way to salvage degrees of freedom if those constraints can be justified.
2.5.2 Missing data
To specify how missing data are handled, we can use missing = 'listwise' to choose to delete individuals with missing data. The better more common approach is to use missing = 'ML' to use full information maximum likelihood (FIML, also known just as ML; this is distinct from restricted maximum likelihood [ReML]).
Never use listwise deletion when you can use ML. ML estimation protects you from bias in more situations. See this paper for more information.
2.5.3 Useful info in plots
We can add useful info to these plots:
semPlot::semPaths(slightly_complicated_sem_fit, what = 'est', layout = 'spring')
semPlot::semPaths(slightly_complicated_sem_fit, what = 'std', layout = 'spring')
2.5.4 Standardized coefficients
We can also get standardized paths from lavaan.
summary(slightly_complicated_sem_fit, standardized = TRUE)## lavaan 0.6-7 ended normally after 30 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 9
##
## Used Total
## Number of observations 820 964
##
## Model Test User Model:
##
## Test statistic 0.096
## Degrees of freedom 1
## P-value (Chi-square) 0.757
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## VS.2 ~
## EXT.1 -0.163 0.079 -2.074 0.038 -0.163 -0.088
## EXT.2 0.478 0.083 5.773 0.000 0.478 0.246
## VS.3 ~
## VS.2 0.513 0.029 17.505 0.000 0.513 0.521
## EXT.1 0.080 0.054 1.470 0.141 0.080 0.044
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## EXT.1 ~~
## EXT.2 0.389 0.027 14.679 0.000 0.389 0.597
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .VS.2 2.246 0.111 20.248 0.000 2.246 0.958
## .VS.3 1.647 0.081 20.248 0.000 1.647 0.724
## EXT.1 0.685 0.034 20.248 0.000 0.685 1.000
## EXT.2 0.621 0.031 20.248 0.000 0.621 1.000The Std.all column isi usually what you want to look at. Covariances in this column are equivalent to correlations (or partial correlations, if they are covariances between residuals). The Std.lv column shows values standardized only on the latent variables. We don’t have any so they are equivalent to the Estimates.
3 Autoregressive-type Models
3.1 Autoregressive Cross-Lag Panel Models
Now we’re starting to get into the longitudinalness of it all.
Remember the AR part of the model is just a series of regressions:
ARCL <- '
y4 ~ y3
y3 ~ y2
y2 ~ y1'When we add in a second variable, we add in that variable’s AR model,
ARCL <- '
#y
y4 ~ y3
y3 ~ y2
y2 ~ y1
#x
x4 ~ x3
x3 ~ x2
x2 ~ x1'and we can also now add in the cross-lag part (\(x_t = y_{t-1} +\dots\)),
ARCL <- '
#y
y4 ~ y3 + x3
y3 ~ y2 + x2
y2 ~ y1 + x1
#x
x4 ~ x3 + y3
x3 ~ x2 + y2
x2 ~ x1 + y1'as well as the contemporaneous (residual) correlations.
ARCL <- '
#y
y4 ~ y3 + x3
y3 ~ y2 + x2
y2 ~ y1 + x1
#x
x4 ~ x3 + y3
x3 ~ x2 + y2
x2 ~ x1 + y1
#x ~~ y
x3 ~~ y3
x2 ~~ y2
x1 ~~ y1'Let’s look at this for the variables in the sample data:
ARCL_model <- '
# Regressions for VS Activation
VS.2 ~ VS.1 + EXT.1
VS.3 ~ VS.2 + EXT.2
VS.4 ~ VS.3 + EXT.3
VS.5 ~ VS.4 + EXT.4
# Regressions for Externalizing Behavior
EXT.2 ~ EXT.1 + VS.1
EXT.3 ~ EXT.2 + VS.2
EXT.4 ~ EXT.3 + VS.3
EXT.5 ~ EXT.4 + VS.4
# Within-Time (Residual) Correlations
VS.1 ~~ EXT.1
VS.2 ~~ EXT.2
VS.3 ~~ EXT.3
VS.4 ~~ EXT.4
VS.5 ~~ EXT.5'
ARCL_fit <- sem(ARCL_model, data = ABCD, meanstructure = TRUE, missing = 'ML')
summary(ARCL_fit)## lavaan 0.6-7 ended normally after 58 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 41
##
## Number of observations 964
## Number of missing patterns 47
##
## Model Test User Model:
##
## Test statistic 660.590
## Degrees of freedom 24
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.2 ~
## VS.1 0.444 0.033 13.596 0.000
## EXT.1 0.053 0.057 0.916 0.360
## VS.3 ~
## VS.2 0.535 0.028 18.866 0.000
## EXT.2 0.022 0.056 0.404 0.686
## VS.4 ~
## VS.3 0.524 0.030 17.494 0.000
## EXT.3 0.102 0.053 1.912 0.056
## VS.5 ~
## VS.4 0.742 0.025 30.146 0.000
## EXT.4 0.061 0.047 1.297 0.195
## EXT.2 ~
## EXT.1 0.578 0.026 22.435 0.000
## VS.1 0.069 0.015 4.720 0.000
## EXT.3 ~
## EXT.2 0.571 0.033 17.523 0.000
## VS.2 0.001 0.016 0.041 0.967
## EXT.4 ~
## EXT.3 0.593 0.027 21.773 0.000
## VS.3 -0.009 0.015 -0.583 0.560
## EXT.5 ~
## EXT.4 0.910 0.044 20.538 0.000
## VS.4 -0.031 0.024 -1.293 0.196
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## VS.1 ~~
## EXT.1 0.029 0.041 0.702 0.483
## .VS.2 ~~
## .EXT.2 0.123 0.032 3.850 0.000
## .VS.3 ~~
## .EXT.3 0.145 0.033 4.333 0.000
## .VS.4 ~~
## .EXT.4 0.163 0.031 5.334 0.000
## .VS.5 ~~
## .EXT.5 0.050 0.039 1.274 0.203
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.2 3.545 0.247 14.354 0.000
## .VS.3 3.015 0.212 14.198 0.000
## .VS.4 3.072 0.222 13.835 0.000
## .VS.5 1.890 0.188 10.047 0.000
## .EXT.2 0.627 0.110 5.694 0.000
## .EXT.3 1.029 0.122 8.421 0.000
## .EXT.4 1.199 0.113 10.616 0.000
## .EXT.5 0.539 0.181 2.984 0.003
## VS.1 6.239 0.047 132.084 0.000
## EXT.1 2.337 0.028 83.108 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.2 2.088 0.098 21.370 0.000
## .VS.3 1.679 0.080 21.076 0.000
## .VS.4 1.716 0.083 20.793 0.000
## .VS.5 1.209 0.058 20.974 0.000
## .EXT.2 0.411 0.020 21.074 0.000
## .EXT.3 0.541 0.026 20.931 0.000
## .EXT.4 0.416 0.021 20.113 0.000
## .EXT.5 1.030 0.051 20.347 0.000
## VS.1 2.151 0.098 21.955 0.000
## EXT.1 0.740 0.034 21.558 0.000semPaths(ARCL_fit, layout='spring', intercepts=FALSE, what = 'std')
BUT REMEMBER NEVER USE THIS MODEL!
3.2 RI-ARCL (RI-CLPM)
We can use the riclpmr pacakge to generate code for this model using a list of variable names grouped by construct.
See also this blog post.
Notice a few things. First, we’re estimating a mean for each variable. But we are also estimating a factor for each variable. The loadings are all set to 1. Why? In the factor model we saw before, the loadings were allowed to be free because we wanted the model to decide how much variance was due to a common factor versus error variance (this is the function of a measurement model). Here, we start with the assumption that the measurment model is well defined (in fact, we’re assuming that the observations (\(x_{1..3}\) and \(y_{1..3}\)) are all measured perfectly. What is this factor for then? It encodes the theoretical assumption that there is a stable trait for each variable \(x\) and \(y\) that causes the measurements. It’s stable, so it causes them equally over time.
In other words, the latent factors \(\kappa\) and \(\omega\) partial out stable, between-person (trait-like) differences from the observations. What’s left behind are the residuals. We can conceptualize the residuals as the perturbations in the observations that are not explained by trait means. In other words, they are within-person changes unexplained by a stable mean score.
We can give “structure” to the residuals by turning them into factors (\(p_{1..3}\) and \(q_{1..3}\)) which we can then use as variables in an ARCL model that describes the within-person change. Now you can see that each observation is determined by the between person factors (\(\kappa\) and \(\omega\)), and the within-person factors (\(p_{1..3}\) and \(q_{1..3}\)), with all of their loadings fixed to 1.
We still have residual variance left over in \(u_{2,3}\) and \(v_{2,3}\).
This becomes a lot to specify, and the lavaan defaults don’t always play nice with these models. To help estimate these things I (John) wrote a little package to help.
#https://johnflournoy.science/riclpmr/
#devtools::install_github('jflournoy/riclpmr')
library(riclpmr)
variable_groups <- list(
EXT = c('EXT.1','EXT.2', 'EXT.3', 'EXT.4', 'EXT.5'),
VS = c('VS.1','VS.2', 'VS.3', 'VS.4', 'VS.5'))
RIARCL_model <- riclpmr::riclpm_text(var_groups = variable_groups, constrain_over_waves = FALSE)
#check the model code
cat(RIARCL_model)## ri_EXT =~ 1*EXT.1 + 1*EXT.2 + 1*EXT.3 + 1*EXT.4 + 1*EXT.5
## ri_VS =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
## ri_EXT ~~ ri_EXT
## ri_VS ~~ ri_VS
## ri_EXT ~~ ri_VS
## EXT.1 ~ EXT.1_mu*1
## EXT.2 ~ EXT.2_mu*1
## EXT.3 ~ EXT.3_mu*1
## EXT.4 ~ EXT.4_mu*1
## EXT.5 ~ EXT.5_mu*1
## VS.1 ~ VS.1_mu*1
## VS.2 ~ VS.2_mu*1
## VS.3 ~ VS.3_mu*1
## VS.4 ~ VS.4_mu*1
## VS.5 ~ VS.5_mu*1
## lat_EXT1 =~ 1*EXT.1
## lat_EXT2 =~ 1*EXT.2
## lat_EXT3 =~ 1*EXT.3
## lat_EXT4 =~ 1*EXT.4
## lat_EXT5 =~ 1*EXT.5
## lat_VS1 =~ 1*VS.1
## lat_VS2 =~ 1*VS.2
## lat_VS3 =~ 1*VS.3
## lat_VS4 =~ 1*VS.4
## lat_VS5 =~ 1*VS.5
## lat_EXT1 ~~ lat_VS1
## lat_EXT2 ~~ lat_VS2
## lat_EXT3 ~~ lat_VS3
## lat_EXT4 ~~ lat_VS4
## lat_EXT5 ~~ lat_VS5
## lat_EXT2 ~ lat_EXT1 + lat_VS1
## lat_EXT3 ~ lat_EXT2 + lat_VS2
## lat_EXT4 ~ lat_EXT3 + lat_VS3
## lat_EXT5 ~ lat_EXT4 + lat_VS4
## lat_EXT2 ~ lat_EXT5 + lat_VS5
## lat_VS2 ~ lat_EXT1 + lat_VS1
## lat_VS3 ~ lat_EXT2 + lat_VS2
## lat_VS4 ~ lat_EXT3 + lat_VS3
## lat_VS5 ~ lat_EXT4 + lat_VS4
## lat_VS2 ~ lat_EXT5 + lat_VS5
## lat_EXT1 ~~ lat_EXT1
## lat_EXT2 ~~ lat_EXT2
## lat_EXT3 ~~ lat_EXT3
## lat_EXT4 ~~ lat_EXT4
## lat_EXT5 ~~ lat_EXT5
## lat_VS1 ~~ lat_VS1
## lat_VS2 ~~ lat_VS2
## lat_VS3 ~~ lat_VS3
## lat_VS4 ~~ lat_VS4
## lat_VS5 ~~ lat_VS5RIARCL_fit <- riclpmr::lavriclpm(RIARCL_model, data = ABCD, missing = 'ML')
summary(RIARCL_fit)## lavaan 0.6-7 ended normally after 65 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 48
##
## Number of observations 964
## Number of missing patterns 47
##
## Model Test User Model:
##
## Test statistic 131.872
## Degrees of freedom 17
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## ri_EXT =~
## EXT.1 1.000
## EXT.2 1.000
## EXT.3 1.000
## EXT.4 1.000
## EXT.5 1.000
## ri_VS =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## lat_EXT1 =~
## EXT.1 1.000
## lat_EXT2 =~
## EXT.2 1.000
## lat_EXT3 =~
## EXT.3 1.000
## lat_EXT4 =~
## EXT.4 1.000
## lat_EXT5 =~
## EXT.5 1.000
## lat_VS1 =~
## VS.1 1.000
## lat_VS2 =~
## VS.2 1.000
## lat_VS3 =~
## VS.3 1.000
## lat_VS4 =~
## VS.4 1.000
## lat_VS5 =~
## VS.5 1.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## lat_EXT2 ~
## lat_EXT1 0.034 0.052 0.657 0.511
## lat_VS1 0.034 0.024 1.372 0.170
## lat_EXT3 ~
## lat_EXT2 -0.179 0.088 -2.029 0.042
## lat_VS2 0.005 0.027 0.182 0.856
## lat_EXT4 ~
## lat_EXT3 0.118 0.052 2.260 0.024
## lat_VS3 -0.025 0.031 -0.787 0.431
## lat_EXT5 ~
## lat_EXT4 0.722 0.088 8.244 0.000
## lat_VS4 -0.054 0.034 -1.607 0.108
## lat_EXT2 ~
## lat_EXT5 0.002 0.023 0.102 0.919
## lat_VS5 0.050 0.030 1.664 0.096
## lat_VS2 ~
## lat_EXT1 -0.234 0.105 -2.223 0.026
## lat_VS1 0.048 0.056 0.867 0.386
## lat_VS3 ~
## lat_EXT2 0.018 0.186 0.097 0.922
## lat_VS2 0.199 0.049 4.085 0.000
## lat_VS4 ~
## lat_EXT3 0.149 0.099 1.501 0.133
## lat_VS3 0.202 0.053 3.793 0.000
## lat_VS5 ~
## lat_EXT4 -0.133 0.114 -1.163 0.245
## lat_VS4 0.541 0.037 14.585 0.000
## lat_VS2 ~
## lat_EXT5 0.085 0.048 1.746 0.081
## lat_VS5 0.215 0.051 4.208 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## ri_EXT ~~
## ri_VS 0.131 0.034 3.821 0.000
## lat_EXT1 ~~
## lat_VS1 -0.095 0.033 -2.863 0.004
## .lat_EXT2 ~~
## .lat_VS2 0.068 0.034 2.012 0.044
## .lat_EXT3 ~~
## .lat_VS3 0.128 0.047 2.748 0.006
## .lat_EXT4 ~~
## .lat_VS4 0.126 0.030 4.144 0.000
## .lat_EXT5 ~~
## .lat_VS5 0.004 0.038 0.100 0.921
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .EXT.1 (EXT.1) 2.337 0.029 81.726 0.000
## .EXT.2 (EXT.2) 2.412 0.026 91.185 0.000
## .EXT.3 (EXT.3) 2.405 0.029 82.313 0.000
## .EXT.4 (EXT.4) 2.541 0.028 91.311 0.000
## .EXT.5 (EXT.5) 2.640 0.043 61.631 0.000
## .VS.1 (VS.1) 6.239 0.048 128.861 0.000
## .VS.2 (VS.2) 6.442 0.052 123.567 0.000
## .VS.3 (VS.3) 6.527 0.050 131.635 0.000
## .VS.4 (VS.4) 6.730 0.052 128.295 0.000
## .VS.5 (VS.5) 7.041 0.052 136.716 0.000
## ri_EXT 0.000
## ri_VS 0.000
## l_EXT1 0.000
## .l_EXT2 0.000
## .l_EXT3 0.000
## .l_EXT4 0.000
## .l_EXT5 0.000
## lt_VS1 0.000
## .lt_VS2 0.000
## .lt_VS3 0.000
## .lt_VS4 0.000
## .lt_VS5 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## ri_EXT 0.417 0.024 17.066 0.000
## ri_VS 0.930 0.076 12.246 0.000
## lat_EXT1 0.351 0.025 14.193 0.000
## .lat_EXT2 0.228 0.022 10.235 0.000
## .lat_EXT3 0.352 0.033 10.609 0.000
## .lat_EXT4 0.263 0.018 14.270 0.000
## .lat_EXT5 1.010 0.050 20.086 0.000
## lat_VS1 1.330 0.085 15.600 0.000
## .lat_VS2 1.490 0.095 15.617 0.000
## .lat_VS3 1.279 0.082 15.613 0.000
## .lat_VS4 1.478 0.087 17.031 0.000
## .lat_VS5 1.019 0.055 18.464 0.000
## .EXT.1 0.000
## .EXT.2 0.000
## .EXT.3 0.000
## .EXT.4 0.000
## .EXT.5 0.000
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000Note: This model centers the between-person differences at 0, so above the means for the variables ri_EXT and ri_VS are both 0. We can use a slightly different model, called the LCM-SR (see below) to create a mean-growth model that is better at describing mean change.
4 Mean change using latent variables
4.1 Intercept and slope latent curve model
Set aside the auto-regressive cross-lag model for a minute, but keep in mind the latent intercept we saw in the RI-CLPM. As is the case in a multilevel growth model, we can describe a series of observations as being a function of an intercept and a slope (with each person getting their own values for these, as in the “random slopes random intercepts” model).
We start by specifying the intercept, very much like in the RI-CLPM:
Int =~ 1*y1 + 1*y2 + 1*y3 + 1*y4Each person gets a value for Int that contributes equally to each of the observed y.
We continue by adding a slope:
int =~ 1*y1 + 1*y2 + 1*y3 + 1*y4
slp =~ 0*y1 + 1*y2 + 2*y3 + 3*y4The slope latent variable increases its influence on the observed y more at each wave. The loadings (0,1,2,3) are essentially the TIME variable in a multilevel model. When the loading for slope is 0, the only contribution to the observed variables is the intercept. This is the timepoint for which the mean of the intercept factor is the expected mean of the data. In other words, time is “centered” at whatever timepoint you set the slope factor to 0.
Let’s look at this using the data we have for externalizing:
# Linear Slope: Externalizing Behavior
LCM_VSlin_model = 'int =~ 1*EXT.1 + 1*EXT.2 + 1*EXT.3 + 1*EXT.4 + 1*EXT.5
slp =~ 0*EXT.1 + 1*EXT.2 + 2*EXT.3 + 3*EXT.4 + 4*EXT.5'
LCM_VSlin_fit = growth(LCM_VSlin_model, data=ABCD, missing = 'ML')## Warning in lav_data_full(data = data, group = group, cluster = cluster, : lavaan WARNING: some cases are empty and will be ignored:
## 649 678summary(LCM_VSlin_fit)## lavaan 0.6-7 ended normally after 29 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 10
##
## Used Total
## Number of observations 962 964
## Number of missing patterns 13
##
## Model Test User Model:
##
## Test statistic 23.630
## Degrees of freedom 10
## P-value (Chi-square) 0.009
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## EXT.1 1.000
## EXT.2 1.000
## EXT.3 1.000
## EXT.4 1.000
## EXT.5 1.000
## slp =~
## EXT.1 0.000
## EXT.2 1.000
## EXT.3 2.000
## EXT.4 3.000
## EXT.5 4.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp -0.016 0.009 -1.928 0.054
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .EXT.1 0.000
## .EXT.2 0.000
## .EXT.3 0.000
## .EXT.4 0.000
## .EXT.5 0.000
## int 2.329 0.025 91.329 0.000
## slp 0.068 0.008 8.170 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .EXT.1 0.288 0.024 12.083 0.000
## .EXT.2 0.256 0.016 15.681 0.000
## .EXT.3 0.370 0.021 17.566 0.000
## .EXT.4 0.142 0.015 9.487 0.000
## .EXT.5 0.881 0.051 17.439 0.000
## int 0.430 0.031 13.754 0.000
## slp 0.024 0.003 7.048 0.000Notice we use the function growth now. This sets lavaa defaults to be appropriate for this (very common) model.
4.1.1 Plotting
We can plot each individual’s trajectory.
pred_data <- as.data.frame(lavPredict(LCM_VSlin_fit, type = 'yhat')) %>%
mutate(id = 1:n()) %>%
pivot_longer(cols = 1:5) %>%
extract(name, c('var', 'wave'), '(EXT)\\.(\\d+)')
ggplot(pred_data) +
geom_line(aes(x = wave, y = value, group = id), alpha = .1)## Warning: Removed 10 row(s) containing missing values (geom_path).
4.2 Constrained latent growth models
The model above corresponds to a random intercept, random slope model. Let’s play around with intercept-only models, and fixed intercepts and slopes.
Intercept only:
# Intercept-Only Model: VS Activation
LCM_VSint = '
int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5'
LCM_VSint_fit = growth(LCM_VSint, data = ABCD, missing = 'ML')
summary(LCM_VSint_fit)## lavaan 0.6-7 ended normally after 29 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 7
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 506.230
## Degrees of freedom 13
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.616 0.040 164.348 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.772 0.095 18.650 0.000
## .VS.2 1.383 0.078 17.753 0.000
## .VS.3 1.054 0.063 16.626 0.000
## .VS.4 1.063 0.066 16.212 0.000
## .VS.5 1.167 0.071 16.457 0.000
## int 1.237 0.070 17.574 0.000semPaths(LCM_VSint_fit, intercepts=TRUE)
pred_data <- as.data.frame(lavPredict(LCM_VSint_fit, type = 'yhat')) %>%
mutate(id = 1:n()) %>%
pivot_longer(cols = 1:5) %>%
extract(name, c('var', 'wave'), '(VS)\\.(\\d+)')
ggplot(pred_data) +
geom_line(aes(x = wave, y = value, group = id), alpha = .1)
Slope variable, but slope is fixed:
# Linear Slope: VS Activation (fixed effect of time)
LCM_VSlin_fixed = '
int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5
slp ~~ 0*slp
slp ~~ 0*int #notice we also have to constrain this!'
LCM_VSlin_fixed_fit = growth(LCM_VSlin_fixed, data=ABCD, missing='ML')
summary(LCM_VSlin_fixed_fit)## lavaan 0.6-7 ended normally after 33 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 8
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 247.515
## Degrees of freedom 12
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.204 0.047 131.243 0.000
## slp 0.195 0.012 16.686 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## slp 0.000
## .VS.1 1.626 0.087 18.633 0.000
## .VS.2 1.364 0.076 17.956 0.000
## .VS.3 1.059 0.063 16.916 0.000
## .VS.4 1.026 0.062 16.467 0.000
## .VS.5 0.917 0.058 15.867 0.000
## int 1.262 0.071 17.827 0.000pred_data <- as.data.frame(lavPredict(LCM_VSlin_fixed_fit, type = 'yhat')) %>%
mutate(id = 1:n()) %>%
pivot_longer(cols = 1:5) %>%
extract(name, c('var', 'wave'), '(VS)\\.(\\d+)')
ggplot(pred_data) +
geom_line(aes(x = wave, y = value, group = id), alpha = .1)
Fixed intercept, slope random:
# Linear Slope: VS Activation (fixed effect of intercept)
LCM_VSlin_fixedint = '
int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5
int ~~ 0*int
int ~~ 0*slp'
LCM_VSlin_fixedint_fit = growth(LCM_VSlin_fixedint, data=ABCD, missing='ML')
summary(LCM_VSlin_fixedint_fit)## lavaan 0.6-7 ended normally after 43 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 8
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 467.927
## Degrees of freedom 12
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.179 0.034 184.223 0.000
## slp 0.215 0.016 13.175 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## int 0.000
## .VS.1 2.155 0.098 21.936 0.000
## .VS.2 2.000 0.094 21.200 0.000
## .VS.3 1.419 0.071 20.071 0.000
## .VS.4 1.084 0.066 16.483 0.000
## .VS.5 0.090 0.060 1.497 0.134
## slp 0.161 0.008 19.169 0.000pred_data <- as.data.frame(lavPredict(LCM_VSlin_fixedint_fit, type = 'yhat')) %>%
mutate(id = 1:n()) %>%
pivot_longer(cols = 1:5) %>%
extract(name, c('var', 'wave'), '(VS)\\.(\\d+)')
ggplot(pred_data) +
geom_line(aes(x = wave, y = value, group = id), alpha = .1)
Random intercept and slope:
# Linear Slope: VS Activation (random effect of time)
LCM_VSlin = '
int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5'
LCM_VSlin_fit = growth(LCM_VSlin, data=ABCD, missing='ML')
summary(LCM_VSlin_fit)## lavaan 0.6-7 ended normally after 41 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 10
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 38.635
## Degrees of freedom 10
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp -0.012 0.022 -0.538 0.591
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.198 0.042 148.400 0.000
## slp 0.204 0.013 15.095 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.257 0.086 14.604 0.000
## .VS.2 1.359 0.076 17.864 0.000
## .VS.3 1.105 0.061 18.014 0.000
## .VS.4 0.942 0.057 16.648 0.000
## .VS.5 0.342 0.058 5.922 0.000
## int 0.948 0.082 11.488 0.000
## slp 0.087 0.010 9.025 0.000semPaths(LCM_VSlin_fit, intercepts=TRUE, edge.color='black', what = 'est')
pred_data <- as.data.frame(lavPredict(LCM_VSlin_fit, type = 'yhat')) %>%
mutate(id = 1:n()) %>%
pivot_longer(cols = 1:5) %>%
extract(name, c('var', 'wave'), '(VS)\\.(\\d+)')
ggplot(pred_data) +
geom_line(aes(x = wave, y = value, group = id), alpha = .1)
Alternative time coding so that intercept is the mean level at the last wave:
# Linear Slope: VS Activation (alternative time coding)
LCM_VSlin2 = '
int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
slp =~ -4*VS.1 + -3*VS.2 + -2*VS.3 + -1*VS.4 + 0*VS.5'
LCM_VSlin2_fit = growth(LCM_VSlin2, data=ABCD, missing='ML')
summary(LCM_VSlin2_fit)## lavaan 0.6-7 ended normally after 41 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 10
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 38.635
## Degrees of freedom 10
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 -4.000
## VS.2 -3.000
## VS.3 -2.000
## VS.4 -1.000
## VS.5 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp 0.335 0.030 11.140 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 7.013 0.053 133.242 0.000
## slp 0.204 0.013 15.095 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.257 0.086 14.604 0.000
## .VS.2 1.359 0.076 17.864 0.000
## .VS.3 1.105 0.061 18.014 0.000
## .VS.4 0.942 0.057 16.648 0.000
## .VS.5 0.342 0.058 5.922 0.000
## int 2.240 0.129 17.385 0.000
## slp 0.087 0.010 9.025 0.000pred_data <- as.data.frame(lavPredict(LCM_VSlin2_fit, type = 'yhat')) %>%
mutate(id = 1:n()) %>%
pivot_longer(cols = 1:5) %>%
extract(name, c('var', 'wave'), '(VS)\\.(\\d+)')
ggplot(pred_data) +
geom_line(aes(x = wave, y = value, group = id), alpha = .1)
5 Advanced latent growth models
5.1 Unconditional Quadratic Latent Growth Curve Model
# Quadratic Effect: VS Activation
LCM.VSqud = 'int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5
qud =~ 0*VS.1 + 1*VS.2 + 4*VS.3 + 9*VS.4 + 16*VS.5'
LCM.VSqud.fit = growth(LCM.VSqud, data=ABCD, missing='ML')
summary(LCM.VSqud.fit)## lavaan 0.6-7 ended normally after 64 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 14
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 26.539
## Degrees of freedom 6
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
## qud =~
## VS.1 0.000
## VS.2 1.000
## VS.3 4.000
## VS.4 9.000
## VS.5 16.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp 0.023 0.118 0.193 0.847
## qud -0.005 0.022 -0.218 0.827
## slp ~~
## qud 0.000 0.023 0.003 0.998
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.266 0.046 135.200 0.000
## slp 0.081 0.039 2.097 0.036
## qud 0.028 0.008 3.389 0.001
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.305 0.153 8.526 0.000
## .VS.2 1.345 0.080 16.896 0.000
## .VS.3 1.084 0.070 15.523 0.000
## .VS.4 0.966 0.066 14.564 0.000
## .VS.5 0.243 0.123 1.974 0.048
## int 0.912 0.147 6.188 0.000
## slp 0.063 0.113 0.558 0.577
## qud 0.002 0.005 0.292 0.770semPaths(LCM.VSqud.fit, intercepts=FALSE, edge.color='black')
# Quadratic Effect: Externalizing Behavior
LCM.EXTqud = 'int =~ 1*EXT.1 + 1*EXT.2 + 1*EXT.3 + 1*EXT.4 + 1*EXT.5
slp =~ 0*EXT.1 + 1*EXT.2 + 2*EXT.3 + 3*EXT.4 + 4*EXT.5
qud =~ 0*EXT.1 + 1*EXT.2 + 4*EXT.3 + 9*EXT.4 + 16*EXT.5'
LCM.EXTqud.fit = growth(LCM.EXTqud, data=ABCD, missing='ML')## Warning in lav_data_full(data = data, group = group, cluster = cluster, : lavaan WARNING: some cases are empty and will be ignored:
## 649 678## Warning in lav_object_post_check(object): lavaan WARNING: some estimated lv
## variances are negativesummary(LCM.EXTqud.fit)## lavaan 0.6-7 ended normally after 51 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 14
##
## Used Total
## Number of observations 962 964
## Number of missing patterns 13
##
## Model Test User Model:
##
## Test statistic 12.606
## Degrees of freedom 6
## P-value (Chi-square) 0.050
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## EXT.1 1.000
## EXT.2 1.000
## EXT.3 1.000
## EXT.4 1.000
## EXT.5 1.000
## slp =~
## EXT.1 0.000
## EXT.2 1.000
## EXT.3 2.000
## EXT.4 3.000
## EXT.5 4.000
## qud =~
## EXT.1 0.000
## EXT.2 1.000
## EXT.3 4.000
## EXT.4 9.000
## EXT.5 16.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp -0.041 0.039 -1.050 0.294
## qud 0.007 0.009 0.852 0.394
## slp ~~
## qud 0.005 0.009 0.595 0.552
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .EXT.1 0.000
## .EXT.2 0.000
## .EXT.3 0.000
## .EXT.4 0.000
## .EXT.5 0.000
## int 2.347 0.028 85.281 0.000
## slp 0.026 0.023 1.162 0.245
## qud 0.012 0.006 1.959 0.050
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .EXT.1 0.286 0.041 6.914 0.000
## .EXT.2 0.264 0.018 14.639 0.000
## .EXT.3 0.374 0.022 17.100 0.000
## .EXT.4 0.143 0.016 9.148 0.000
## .EXT.5 0.887 0.066 13.472 0.000
## int 0.454 0.047 9.648 0.000
## slp 0.001 0.039 0.036 0.971
## qud -0.001 0.002 -0.533 0.594LCM.EXTqud2 = 'int =~ 1*EXT.1 + 1*EXT.2 + 1*EXT.3 + 1*EXT.4 + 1*EXT.5
slp =~ 0*EXT.1 + 1*EXT.2 + 2*EXT.3 + 3*EXT.4 + 4*EXT.5
qud =~ 0*EXT.1 + 1*EXT.2 + 4*EXT.3 + 9*EXT.4 + 16*EXT.5
qud ~~ 0*qud
qud ~~ 0*int + 0*slp'
LCM.EXTqud2.fit = growth(LCM.EXTqud2, data=ABCD, missing='ML')## Warning in lav_data_full(data = data, group = group, cluster = cluster, : lavaan WARNING: some cases are empty and will be ignored:
## 649 678summary(LCM.EXTqud2.fit)## lavaan 0.6-7 ended normally after 36 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 11
##
## Used Total
## Number of observations 962 964
## Number of missing patterns 13
##
## Model Test User Model:
##
## Test statistic 19.153
## Degrees of freedom 9
## P-value (Chi-square) 0.024
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## EXT.1 1.000
## EXT.2 1.000
## EXT.3 1.000
## EXT.4 1.000
## EXT.5 1.000
## slp =~
## EXT.1 0.000
## EXT.2 1.000
## EXT.3 2.000
## EXT.4 3.000
## EXT.5 4.000
## qud =~
## EXT.1 0.000
## EXT.2 1.000
## EXT.3 4.000
## EXT.4 9.000
## EXT.5 16.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## qud 0.000
## slp ~~
## qud 0.000
## int ~~
## slp -0.017 0.009 -2.003 0.045
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .EXT.1 0.000
## .EXT.2 0.000
## .EXT.3 0.000
## .EXT.4 0.000
## .EXT.5 0.000
## int 2.348 0.027 86.745 0.000
## slp 0.023 0.023 0.998 0.318
## qud 0.013 0.006 2.118 0.034
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## qud 0.000
## .EXT.1 0.286 0.024 11.998 0.000
## .EXT.2 0.257 0.016 15.689 0.000
## .EXT.3 0.367 0.021 17.514 0.000
## .EXT.4 0.142 0.015 9.509 0.000
## .EXT.5 0.879 0.050 17.428 0.000
## int 0.432 0.031 13.777 0.000
## slp 0.024 0.003 7.069 0.0005.2 Unconditional Piecewise Latent Growth Curve Model
# Piecewise Linear Effect: VS Activation
LCM.VSpw = 'int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
pw1 =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 3*VS.5
pw2 =~ 0*VS.1 + 0*VS.2 + 0*VS.3 + 1*VS.4 + 2*VS.5'
LCM.VSpw.fit = growth(LCM.VSpw, data=ABCD, missing='ML')## Warning in lav_object_post_check(object): lavaan WARNING: some estimated lv
## variances are negativesummary(LCM.VSpw.fit)## lavaan 0.6-7 ended normally after 52 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 14
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 38.154
## Degrees of freedom 6
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## pw1 =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 3.000
## pw2 =~
## VS.1 0.000
## VS.2 0.000
## VS.3 0.000
## VS.4 1.000
## VS.5 2.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## pw1 0.015 0.050 0.291 0.771
## pw2 -0.025 0.044 -0.561 0.575
## pw1 ~~
## pw2 0.089 0.029 3.028 0.002
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.288 0.045 139.147 0.000
## pw1 0.098 0.022 4.484 0.000
## pw2 0.223 0.026 8.617 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.324 0.114 11.568 0.000
## .VS.2 1.359 0.078 17.480 0.000
## .VS.3 1.164 0.080 14.557 0.000
## .VS.4 0.883 0.070 12.701 0.000
## .VS.5 0.530 0.118 4.486 0.000
## int 0.941 0.110 8.584 0.000
## pw1 0.030 0.030 0.996 0.319
## pw2 -0.061 0.065 -0.937 0.3495.3 Unconditional Latent Basis Growth Curve Model
# Latent Basis Effect: VS Activation
LCM.VSfl = 'int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
fl =~ 0*VS.1 + 1*VS.2 + VS.3 + VS.4 + VS.5'
LCM.VSfl.fit = growth(LCM.VSfl, data=ABCD, missing='ML')
summary(LCM.VSfl.fit)## lavaan 0.6-7 ended normally after 88 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 13
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 18.435
## Degrees of freedom 7
## P-value (Chi-square) 0.010
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## fl =~
## VS.1 0.000
## VS.2 1.000
## VS.3 1.514 0.210 7.194 0.000
## VS.4 2.416 0.341 7.078 0.000
## VS.5 3.948 0.617 6.398 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## fl -0.007 0.022 -0.322 0.747
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.232 0.045 137.239 0.000
## fl 0.206 0.039 5.336 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.277 0.084 15.185 0.000
## .VS.2 1.375 0.077 17.926 0.000
## .VS.3 1.111 0.062 17.955 0.000
## .VS.4 1.015 0.061 16.514 0.000
## .VS.5 0.102 0.101 1.001 0.317
## int 0.942 0.076 12.462 0.000
## fl 0.105 0.035 2.973 0.0035.4 Conditional Latent Growth Curve Model: TICs
# Conditional TIC Model
LCM.TIC = 'int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5
int ~ female + advers
slp ~ female + advers'
LCM.TIC.fit = growth(LCM.TIC, data=ABCD, missing='ML')
summary(LCM.TIC.fit)## lavaan 0.6-7 ended normally after 46 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 14
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 66.459
## Degrees of freedom 16
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## int ~
## female 0.015 0.086 0.176 0.860
## advers 0.034 0.086 0.399 0.690
## slp ~
## female 0.050 0.028 1.799 0.072
## advers 0.015 0.027 0.535 0.593
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .int ~~
## .slp -0.011 0.022 -0.513 0.608
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## .int 6.172 0.078 79.118 0.000
## .slp 0.174 0.025 6.954 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.258 0.086 14.581 0.000
## .VS.2 1.361 0.076 17.847 0.000
## .VS.3 1.107 0.061 18.000 0.000
## .VS.4 0.940 0.057 16.612 0.000
## .VS.5 0.344 0.058 5.950 0.000
## .int 0.946 0.082 11.470 0.000
## .slp 0.086 0.010 8.951 0.000semPaths(LCM.TIC.fit, intercepts=FALSE, edge.color='black')
# Conditional TIC Model with Explicit Exogenous Covariance
LCM.TIC = 'int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5
int ~ female + advers
slp ~ female + advers
female ~~ advers'
LCM.TIC.fit = growth(LCM.TIC, data=ABCD, missing='ML')
summary(LCM.TIC.fit)## lavaan 0.6-7 ended normally after 45 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 17
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 1273.142
## Degrees of freedom 18
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## int ~
## female 0.015 0.086 0.176 0.860
## advers 0.034 0.086 0.399 0.690
## slp ~
## female 0.050 0.028 1.799 0.072
## advers 0.015 0.027 0.535 0.593
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## female ~~
## advers 0.196 0.017 11.448 0.000
## .int ~~
## .slp -0.011 0.022 -0.513 0.608
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## female 0.000
## advers 0.000
## .int 6.172 0.078 79.118 0.000
## .slp 0.174 0.025 6.954 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.258 0.086 14.581 0.000
## .VS.2 1.361 0.076 17.847 0.000
## .VS.3 1.107 0.061 18.000 0.000
## .VS.4 0.940 0.057 16.612 0.000
## .VS.5 0.344 0.058 5.950 0.000
## female 0.446 0.020 21.954 0.000
## advers 0.548 0.025 21.954 0.000
## .int 0.946 0.082 11.470 0.000
## .slp 0.086 0.010 8.951 0.0005.5 Conditional Latent Growth Curve Model: TVCs
# Conditional TVC Model
LCM.TVC1 = 'int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5
VS.1 ~ EXT.1
VS.2 ~ EXT.2
VS.3 ~ EXT.3
VS.4 ~ EXT.4
VS.5 ~ EXT.5
int ~~ EXT.1 + EXT.2 + EXT.3 + EXT.4 + EXT.5
slp ~~ EXT.1 + EXT.2 + EXT.3 + EXT.4 + EXT.5'
LCM.TVC1.fit = growth(LCM.TVC1, data=ABCD, missing='ML')
summary(LCM.TVC1.fit)## lavaan 0.6-7 ended normally after 162 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 30
##
## Number of observations 964
## Number of missing patterns 47
##
## Model Test User Model:
##
## Test statistic 11176.284
## Degrees of freedom 35
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.1 ~
## EXT.1 0.274 0.056 4.869 0.000
## VS.2 ~
## EXT.2 0.240 0.041 5.897 0.000
## VS.3 ~
## EXT.3 0.167 0.032 5.249 0.000
## VS.4 ~
## EXT.4 0.147 0.030 4.857 0.000
## VS.5 ~
## EXT.5 0.150 0.037 4.051 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## EXT.1 -2.051 0.421 -4.873 0.000
## EXT.2 1.097 0.384 2.856 0.004
## EXT.3 -0.200 0.374 -0.535 0.593
## EXT.4 -1.209 0.425 -2.843 0.004
## EXT.5 1.605 0.351 4.572 0.000
## slp ~~
## EXT.1 0.738 0.135 5.465 0.000
## EXT.2 0.134 0.146 0.919 0.358
## EXT.3 0.167 0.113 1.473 0.141
## EXT.4 0.278 0.137 2.021 0.043
## EXT.5 -0.702 0.143 -4.901 0.000
## int ~~
## slp -0.400 0.100 -3.994 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## EXT.1 0.000
## EXT.2 0.000
## EXT.3 0.000
## EXT.4 0.000
## EXT.5 0.000
## int 5.980 0.133 45.107 0.000
## slp -0.008 0.044 -0.179 0.858
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.208 0.085 14.238 0.000
## .VS.2 1.320 0.074 17.838 0.000
## .VS.3 1.070 0.060 17.694 0.000
## .VS.4 0.910 0.055 16.429 0.000
## .VS.5 0.321 0.056 5.739 0.000
## EXT.1 6.227 0.289 21.514 0.000
## EXT.2 6.596 0.309 21.342 0.000
## EXT.3 6.735 0.322 20.909 0.000
## EXT.4 7.474 0.368 20.318 0.000
## EXT.5 8.939 0.442 20.236 0.000
## int 2.200 0.348 6.324 0.000
## slp 0.240 0.038 6.240 0.000semPaths(LCM.TVC1.fit, intercepts=FALSE, edge.color='black')
# Conditional TVC Model with Equality Constraints
LCM.TVC2 = 'int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5
VS.1 ~ a*EXT.1
VS.2 ~ a*EXT.2
VS.3 ~ a*EXT.3
VS.4 ~ a*EXT.4
VS.5 ~ a*EXT.5
int ~~ EXT.1 + EXT.2 + EXT.3 + EXT.4 + EXT.5
slp ~~ EXT.1 + EXT.2 + EXT.3 + EXT.4 + EXT.5'
LCM.TVC2.fit = growth(LCM.TVC2, data=ABCD, missing='ML')
summary(LCM.TVC2.fit)## lavaan 0.6-7 ended normally after 144 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 30
## Number of equality constraints 4
##
## Number of observations 964
## Number of missing patterns 47
##
## Model Test User Model:
##
## Test statistic 11190.367
## Degrees of freedom 39
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.1 ~
## EXT.1 (a) 0.181 0.026 6.882 0.000
## VS.2 ~
## EXT.2 (a) 0.181 0.026 6.882 0.000
## VS.3 ~
## EXT.3 (a) 0.181 0.026 6.882 0.000
## VS.4 ~
## EXT.4 (a) 0.181 0.026 6.882 0.000
## VS.5 ~
## EXT.5 (a) 0.181 0.026 6.882 0.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## EXT.1 -1.700 0.369 -4.603 0.000
## EXT.2 1.205 0.379 3.179 0.001
## EXT.3 -0.244 0.373 -0.654 0.513
## EXT.4 -1.186 0.429 -2.768 0.006
## EXT.5 1.662 0.349 4.758 0.000
## slp ~~
## EXT.1 0.628 0.118 5.306 0.000
## EXT.2 0.111 0.144 0.771 0.441
## EXT.3 0.175 0.112 1.559 0.119
## EXT.4 0.267 0.136 1.966 0.049
## EXT.5 -0.776 0.127 -6.096 0.000
## int ~~
## slp -0.347 0.089 -3.896 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## EXT.1 0.000
## EXT.2 0.000
## EXT.3 0.000
## EXT.4 0.000
## EXT.5 0.000
## int 5.984 0.132 45.205 0.000
## slp -0.008 0.044 -0.190 0.849
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.206 0.085 14.227 0.000
## .VS.2 1.326 0.074 17.875 0.000
## .VS.3 1.076 0.061 17.683 0.000
## .VS.4 0.909 0.056 16.368 0.000
## .VS.5 0.333 0.056 5.898 0.000
## EXT.1 6.230 0.290 21.500 0.000
## EXT.2 6.596 0.309 21.336 0.000
## EXT.3 6.734 0.322 20.922 0.000
## EXT.4 7.460 0.366 20.356 0.000
## EXT.5 8.938 0.441 20.247 0.000
## int 2.043 0.312 6.543 0.000
## slp 0.226 0.036 6.209 0.000semPaths(LCM.TVC2.fit, intercepts=FALSE, edge.color='black')
5.6 Multivariate Latent Curve Model
mLCM = 'VS.int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
VS.slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5
EXT.int =~ 1*EXT.1 + 1*EXT.2 + 1*EXT.3 + 1*EXT.4 + 1*EXT.5
EXT.slp =~ 0*EXT.1 + 1*EXT.2 + 2*EXT.3 + 3*EXT.4 + 4*EXT.5
VS.1 ~~ EXT.1
VS.2 ~~ EXT.2
VS.3 ~~ EXT.3
VS.4 ~~ EXT.4
VS.5 ~~ EXT.5
VS.int ~~ VS.slp + EXT.int + EXT.slp
VS.slp ~~ EXT.int + EXT.slp
EXT.int ~~ EXT.slp'
mLCM.fit = growth(mLCM, data=ABCD, missing='ML')
summary(mLCM.fit)## lavaan 0.6-7 ended normally after 63 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 29
##
## Number of observations 964
## Number of missing patterns 47
##
## Model Test User Model:
##
## Test statistic 97.489
## Degrees of freedom 36
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## VS.int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## VS.slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
## EXT.int =~
## EXT.1 1.000
## EXT.2 1.000
## EXT.3 1.000
## EXT.4 1.000
## EXT.5 1.000
## EXT.slp =~
## EXT.1 0.000
## EXT.2 1.000
## EXT.3 2.000
## EXT.4 3.000
## EXT.5 4.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 ~~
## .EXT.1 -0.007 0.032 -0.216 0.829
## .VS.2 ~~
## .EXT.2 0.084 0.025 3.344 0.001
## .VS.3 ~~
## .EXT.3 0.123 0.027 4.571 0.000
## .VS.4 ~~
## .EXT.4 0.092 0.020 4.559 0.000
## .VS.5 ~~
## .EXT.5 -0.029 0.036 -0.806 0.420
## VS.int ~~
## VS.slp -0.008 0.021 -0.394 0.694
## EXT.int 0.019 0.036 0.542 0.588
## EXT.slp 0.040 0.012 3.388 0.001
## VS.slp ~~
## EXT.int 0.059 0.012 5.061 0.000
## EXT.slp -0.010 0.004 -2.770 0.006
## EXT.int ~~
## EXT.slp -0.016 0.008 -1.850 0.064
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## .EXT.1 0.000
## .EXT.2 0.000
## .EXT.3 0.000
## .EXT.4 0.000
## .EXT.5 0.000
## VS.int 6.203 0.042 148.937 0.000
## VS.slp 0.201 0.013 14.916 0.000
## EXT.int 2.327 0.025 91.340 0.000
## EXT.slp 0.071 0.008 8.608 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.274 0.087 14.716 0.000
## .VS.2 1.353 0.076 17.885 0.000
## .VS.3 1.112 0.062 18.060 0.000
## .VS.4 0.957 0.057 16.796 0.000
## .VS.5 0.316 0.057 5.597 0.000
## .EXT.1 0.285 0.023 12.235 0.000
## .EXT.2 0.256 0.016 15.725 0.000
## .EXT.3 0.376 0.021 17.624 0.000
## .EXT.4 0.141 0.015 9.496 0.000
## .EXT.5 0.876 0.050 17.502 0.000
## VS.int 0.937 0.082 11.434 0.000
## VS.slp 0.087 0.010 9.102 0.000
## EXT.int 0.431 0.031 13.814 0.000
## EXT.slp 0.024 0.003 7.155 0.000semPaths(mLCM.fit, intercepts=FALSE, edge.color='black')
5.7 Multivariate Latent Curve Model with Structured Residuals
mLCMSR = '# Define the Latent Factors
VS.int =~ 1*VS.1 + 1*VS.2 + 1*VS.3 + 1*VS.4 + 1*VS.5
VS.slp =~ 0*VS.1 + 1*VS.2 + 2*VS.3 + 3*VS.4 + 4*VS.5
EXT.int =~ 1*EXT.1 + 1*EXT.2 + 1*EXT.3 + 1*EXT.4 + 1*EXT.5
EXT.slp =~ 0*EXT.1 + 1*EXT.2 + 2*EXT.3 + 3*EXT.4 + 4*EXT.5
# Factor Covariances
VS.int ~~ VS.slp + EXT.int + EXT.slp
VS.slp ~~ EXT.int + EXT.slp
EXT.int ~~ EXT.slp
# Define Phantom Variables
VS.1 ~~ 0*VS.1; srVS.1 =~ 1*VS.1; srVS.1 ~ 0; srVS.1 ~~ srVS.1
VS.2 ~~ 0*VS.2; srVS.2 =~ 1*VS.2; srVS.2 ~ 0; srVS.2 ~~ srVS.2
VS.3 ~~ 0*VS.3; srVS.3 =~ 1*VS.3; srVS.3 ~ 0; srVS.3 ~~ srVS.3
VS.4 ~~ 0*VS.4; srVS.4 =~ 1*VS.4; srVS.4 ~ 0; srVS.4 ~~ srVS.4
VS.5 ~~ 0*VS.5; srVS.5 =~ 1*VS.5; srVS.5 ~ 0; srVS.5 ~~ srVS.5
EXT.1 ~~ 0*EXT.1; srEXT.1 =~ 1*EXT.1; srEXT.1 ~ 0; srEXT.1 ~~ srEXT.1
EXT.2 ~~ 0*EXT.2; srEXT.2 =~ 1*EXT.2; srEXT.2 ~ 0; srEXT.2 ~~ srEXT.2
EXT.3 ~~ 0*EXT.3; srEXT.3 =~ 1*EXT.3; srEXT.3 ~ 0; srEXT.3 ~~ srEXT.3
EXT.4 ~~ 0*EXT.4; srEXT.4 =~ 1*EXT.4; srEXT.4 ~ 0; srEXT.4 ~~ srEXT.4
EXT.5 ~~ 0*EXT.5; srEXT.5 =~ 1*EXT.5; srEXT.5 ~ 0; srEXT.5 ~~ srEXT.5
# Structured Residuals Regressions + Covariances
srVS.2 ~ a*srVS.1 + b*srEXT.1
srVS.3 ~ a*srVS.2 + b*srEXT.2
srVS.4 ~ a*srVS.3 + b*srEXT.3
srVS.5 ~ a*srVS.4 + b*srEXT.4
srEXT.2 ~ c*srEXT.1 + d*srVS.1
srEXT.3 ~ c*srEXT.2 + d*srVS.2
srEXT.4 ~ c*srEXT.3 + d*srVS.3
srEXT.5 ~ c*srEXT.4 + d*srVS.4
srVS.1 ~~ srEXT.1
srVS.2 ~~ srEXT.2
srVS.3 ~~ srEXT.3
srVS.4 ~~ srEXT.4
srVS.5 ~~ srEXT.5
# Uncouple 1st SRs from Growth Factors
VS.int ~~ 0*srVS.1 + 0*srEXT.1
VS.slp ~~ 0*srVS.1 + 0*srEXT.1
EXT.int ~~ 0*srVS.1 + 0*srEXT.1
EXT.slp ~~ 0*srVS.1 + 0*srEXT.1
'
mLCMSR.fit = growth(mLCMSR, data=ABCD, missing='ML')
summary(mLCMSR.fit)## lavaan 0.6-7 ended normally after 84 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 45
## Number of equality constraints 12
##
## Number of observations 964
## Number of missing patterns 47
##
## Model Test User Model:
##
## Test statistic 87.795
## Degrees of freedom 32
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## VS.int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## VS.slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
## EXT.int =~
## EXT.1 1.000
## EXT.2 1.000
## EXT.3 1.000
## EXT.4 1.000
## EXT.5 1.000
## EXT.slp =~
## EXT.1 0.000
## EXT.2 1.000
## EXT.3 2.000
## EXT.4 3.000
## EXT.5 4.000
## srVS.1 =~
## VS.1 1.000
## srVS.2 =~
## VS.2 1.000
## srVS.3 =~
## VS.3 1.000
## srVS.4 =~
## VS.4 1.000
## srVS.5 =~
## VS.5 1.000
## srEXT.1 =~
## EXT.1 1.000
## srEXT.2 =~
## EXT.2 1.000
## srEXT.3 =~
## EXT.3 1.000
## srEXT.4 =~
## EXT.4 1.000
## srEXT.5 =~
## EXT.5 1.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## srVS.2 ~
## srVS.1 (a) 0.032 0.032 1.012 0.312
## srEXT.1 (b) 0.109 0.054 2.000 0.045
## srVS.3 ~
## srVS.2 (a) 0.032 0.032 1.012 0.312
## srEXT.2 (b) 0.109 0.054 2.000 0.045
## srVS.4 ~
## srVS.3 (a) 0.032 0.032 1.012 0.312
## srEXT.3 (b) 0.109 0.054 2.000 0.045
## srVS.5 ~
## srVS.4 (a) 0.032 0.032 1.012 0.312
## srEXT.4 (b) 0.109 0.054 2.000 0.045
## srEXT.2 ~
## srEXT.1 (c) -0.058 0.034 -1.694 0.090
## srVS.1 (d) 0.019 0.013 1.431 0.152
## srEXT.3 ~
## srEXT.2 (c) -0.058 0.034 -1.694 0.090
## srVS.2 (d) 0.019 0.013 1.431 0.152
## srEXT.4 ~
## srEXT.3 (c) -0.058 0.034 -1.694 0.090
## srVS.3 (d) 0.019 0.013 1.431 0.152
## srEXT.5 ~
## srEXT.4 (c) -0.058 0.034 -1.694 0.090
## srVS.4 (d) 0.019 0.013 1.431 0.152
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## VS.int ~~
## VS.slp 0.008 0.026 0.297 0.767
## EXT.int -0.007 0.039 -0.174 0.862
## EXT.slp 0.048 0.013 3.781 0.000
## VS.slp ~~
## EXT.int 0.066 0.012 5.390 0.000
## EXT.slp -0.014 0.004 -3.360 0.001
## EXT.int ~~
## EXT.slp -0.022 0.009 -2.505 0.012
## srVS.1 ~~
## srEXT.1 0.011 0.034 0.322 0.748
## .srVS.2 ~~
## .srEXT.2 0.108 0.028 3.892 0.000
## .srVS.3 ~~
## .srEXT.3 0.135 0.027 4.927 0.000
## .srVS.4 ~~
## .srEXT.4 0.117 0.023 5.045 0.000
## .srVS.5 ~~
## .srEXT.5 -0.020 0.036 -0.564 0.573
## VS.int ~~
## srVS.1 0.000
## srEXT.1 0.000
## VS.slp ~~
## srVS.1 0.000
## srEXT.1 0.000
## EXT.int ~~
## srVS.1 0.000
## srEXT.1 0.000
## EXT.slp ~~
## srVS.1 0.000
## srEXT.1 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## srVS.1 0.000
## .srVS.2 0.000
## .srVS.3 0.000
## .srVS.4 0.000
## .srVS.5 0.000
## srEXT.1 0.000
## .srEXT.2 0.000
## .srEXT.3 0.000
## .srEXT.4 0.000
## .srEXT.5 0.000
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## .EXT.1 0.000
## .EXT.2 0.000
## .EXT.3 0.000
## .EXT.4 0.000
## .EXT.5 0.000
## VS.int 6.202 0.042 149.172 0.000
## VS.slp 0.201 0.013 14.962 0.000
## EXT.int 2.328 0.025 91.319 0.000
## EXT.slp 0.071 0.008 8.526 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## srVS.1 1.329 0.099 13.382 0.000
## .VS.2 0.000
## .srVS.2 1.388 0.085 16.329 0.000
## .VS.3 0.000
## .srVS.3 1.119 0.062 17.969 0.000
## .VS.4 0.000
## .srVS.4 1.003 0.071 14.030 0.000
## .VS.5 0.000
## .srVS.5 0.346 0.061 5.623 0.000
## .EXT.1 0.000
## srEXT.1 0.266 0.023 11.479 0.000
## .EXT.2 0.000
## .srEXT.2 0.245 0.017 14.256 0.000
## .EXT.3 0.000
## .srEXT.3 0.357 0.023 15.463 0.000
## .EXT.4 0.000
## .srEXT.4 0.129 0.017 7.801 0.000
## .EXT.5 0.000
## .srEXT.5 0.874 0.050 17.467 0.000
## VS.int 0.878 0.097 9.030 0.000
## VS.slp 0.081 0.011 7.278 0.000
## EXT.int 0.452 0.032 13.958 0.000
## EXT.slp 0.028 0.004 7.312 0.000semPaths(mLCMSR.fit, layout='spring', intercepts = F, residuals = F)
5.8 Two-Timepoint Latent Change Score
# FIML 2TP LCS Model
LCS1 = '# Set Regression Path to 1
VS.2 ~ 1*VS.1
# Define Change Latent Variable
dVS.21 =~ 1*VS.2
dVS.21 ~ 1
# Estimate Intercept and Variance of V.1
VS.1 ~ 1
VS.1 ~~ VS.1
# Constraint Intercept and Variance of V.2 to 0
VS.2 ~ 0
VS.2 ~~ 0*VS.2
'
LCS1.fit = sem(LCS1, data=ABCD, missing='ML')
summary(LCS1.fit)## lavaan 0.6-7 ended normally after 15 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 4
##
## Number of observations 964
## Number of missing patterns 2
##
## Model Test User Model:
##
## Test statistic 240.497
## Degrees of freedom 1
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## dVS.21 =~
## VS.2 1.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.2 ~
## VS.1 1.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## dVS.21 0.188 0.054 3.460 0.001
## VS.1 6.239 0.047 132.084 0.000
## .VS.2 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## VS.1 2.151 0.098 21.954 0.000
## .VS.2 0.000
## dVS.21 2.708 0.126 21.413 0.000# Complete Case 2TP LCS Model
LCS1.fit2 = sem(LCS1, data=ABCD, missing='listwise')
summary(LCS1.fit2)## lavaan 0.6-7 ended normally after 13 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 4
##
## Used Total
## Number of observations 917 964
##
## Model Test User Model:
##
## Test statistic 240.497
## Degrees of freedom 1
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## dVS.21 =~
## VS.2 1.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.2 ~
## VS.1 1.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## dVS.21 0.188 0.054 3.460 0.001
## VS.1 6.248 0.048 130.781 0.000
## .VS.2 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## VS.1 2.093 0.098 21.413 0.000
## .VS.2 0.000
## dVS.21 2.708 0.126 21.413 0.000# Paired Samples T-Test
t.test(ABCD$VS.1, ABCD$VS.2, paired=TRUE)##
## Paired t-test
##
## data: ABCD$VS.1 and ABCD$VS.2
## t = -3.4577, df = 916, p-value = 0.00057
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.29468244 -0.08128474
## sample estimates:
## mean of the differences
## -0.1879836describe(ABCD[,c('VS.1','VS.2')])## vars n mean sd median trimmed mad min max range skew kurtosis se
## VS.1 1 964 6.24 1.47 6.19 6.22 1.41 1.43 10 8.57 0.09 0.09 0.05
## VS.2 2 917 6.44 1.59 6.67 6.41 1.41 1.90 10 8.10 0.10 -0.24 0.056.248 + .188## [1] 6.436# Proportional Change LCS
LCS2 = '# Set Regression Path to 1
VS.2 ~ 1*VS.1
# Define Change Latent Variable
dVS.21 =~ 1*VS.2
dVS.21 ~ 1
# Regress Change on Initial Status
dVS.21 ~ VS.1
# Estimate Intercept and Variance of V.1
VS.1 ~ 1
VS.1 ~~ VS.1
# Constraint Intercept and Variance of V.2 to 0
VS.2 ~ 0
VS.2 ~~ 0*VS.2
'
LCS2.fit = sem(LCS2, data=ABCD, missing='ML')
summary(LCS2.fit)## lavaan 0.6-7 ended normally after 29 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 5
##
## Number of observations 964
## Number of missing patterns 2
##
## Model Test User Model:
##
## Test statistic 0.000
## Degrees of freedom 0
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## dVS.21 =~
## VS.2 1.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## VS.2 ~
## VS.1 1.000
## dVS.21 ~
## VS.1 -0.546 0.033 -16.583 0.000
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .dVS.21 3.601 0.211 17.045 0.000
## VS.1 6.239 0.047 132.084 0.000
## .VS.2 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## VS.1 2.151 0.098 21.954 0.000
## .VS.2 0.000
## .dVS.21 2.083 0.097 21.413 0.0005.9 Latent Change Score Trajectory Model
# LCS Trajectory Model
LCSt = '# Define Phantom Variables
pVS.1 =~ 1*VS.1; VS.1 ~ 0; VS.1 ~~ VS.1; pVS.1 ~~ 0*pVS.1
pVS.2 =~ 1*VS.2; VS.2 ~ 0; VS.2 ~~ VS.2; pVS.2 ~~ 0*pVS.2
pVS.3 =~ 1*VS.3; VS.3 ~ 0; VS.3 ~~ VS.3; pVS.3 ~~ 0*pVS.3
pVS.4 =~ 1*VS.4; VS.4 ~ 0; VS.4 ~~ VS.4; pVS.4 ~~ 0*pVS.4
pVS.5 =~ 1*VS.5; VS.5 ~ 0; VS.5 ~~ VS.5; pVS.5 ~~ 0*pVS.5
# Regressions Between Adjacent Observations
pVS.2 ~ 1*pVS.1
pVS.3 ~ 1*pVS.2
pVS.4 ~ 1*pVS.3
pVS.5 ~ 1*pVS.4
# Define Change Latent Variables
dVS.21 =~ 1*pVS.2; dVS.21 ~~ 0*dVS.21
dVS.32 =~ 1*pVS.3; dVS.32 ~~ 0*dVS.32
dVS.43 =~ 1*pVS.4; dVS.43 ~~ 0*dVS.43
dVS.54 =~ 1*pVS.5; dVS.54 ~~ 0*dVS.54
# Define Intercept and Slope
int =~ 1*pVS.1
slp =~ 1*dVS.21 + 1*dVS.32 + 1*dVS.43 + 1*dVS.54
int ~ 1
slp ~ 1
int ~~ int + slp
slp ~~ slp
'
LCSt.fit = sem(LCSt, data=ABCD, missing='ML')
summary(LCSt.fit)## lavaan 0.6-7 ended normally after 41 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 10
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 38.635
## Degrees of freedom 10
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## pVS.1 =~
## VS.1 1.000
## pVS.2 =~
## VS.2 1.000
## pVS.3 =~
## VS.3 1.000
## pVS.4 =~
## VS.4 1.000
## pVS.5 =~
## VS.5 1.000
## dVS.21 =~
## pVS.2 1.000
## dVS.32 =~
## pVS.3 1.000
## dVS.43 =~
## pVS.4 1.000
## dVS.54 =~
## pVS.5 1.000
## int =~
## pVS.1 1.000
## slp =~
## dVS.21 1.000
## dVS.32 1.000
## dVS.43 1.000
## dVS.54 1.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## pVS.2 ~
## pVS.1 1.000
## pVS.3 ~
## pVS.2 1.000
## pVS.4 ~
## pVS.3 1.000
## pVS.5 ~
## pVS.4 1.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp -0.012 0.022 -0.538 0.591
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.198 0.042 148.400 0.000
## slp 0.204 0.013 15.095 0.000
## .pVS.1 0.000
## .pVS.2 0.000
## .pVS.3 0.000
## .pVS.4 0.000
## .pVS.5 0.000
## .dVS.21 0.000
## .dVS.32 0.000
## .dVS.43 0.000
## .dVS.54 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.257 0.086 14.604 0.000
## .pVS.1 0.000
## .VS.2 1.359 0.076 17.864 0.000
## .pVS.2 0.000
## .VS.3 1.105 0.061 18.014 0.000
## .pVS.3 0.000
## .VS.4 0.942 0.057 16.648 0.000
## .pVS.4 0.000
## .VS.5 0.342 0.058 5.922 0.000
## .pVS.5 0.000
## .dVS.21 0.000
## .dVS.32 0.000
## .dVS.43 0.000
## .dVS.54 0.000
## int 0.948 0.082 11.488 0.000
## slp 0.087 0.010 9.025 0.000# Identical LCM Model
summary(LCM_VSlin_fit)## lavaan 0.6-7 ended normally after 41 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 10
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 38.635
## Degrees of freedom 10
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## int =~
## VS.1 1.000
## VS.2 1.000
## VS.3 1.000
## VS.4 1.000
## VS.5 1.000
## slp =~
## VS.1 0.000
## VS.2 1.000
## VS.3 2.000
## VS.4 3.000
## VS.5 4.000
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp -0.012 0.022 -0.538 0.591
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.198 0.042 148.400 0.000
## slp 0.204 0.013 15.095 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.257 0.086 14.604 0.000
## .VS.2 1.359 0.076 17.864 0.000
## .VS.3 1.105 0.061 18.014 0.000
## .VS.4 0.942 0.057 16.648 0.000
## .VS.5 0.342 0.058 5.922 0.000
## int 0.948 0.082 11.488 0.000
## slp 0.087 0.010 9.025 0.000# Proportional Change LCS Trajectory Model
LCSpt = '# Define Phantom Variables
pVS.1 =~ 1*VS.1; VS.1 ~ 0; VS.1 ~~ VS.1; pVS.1 ~~ 0*pVS.1
pVS.2 =~ 1*VS.2; VS.2 ~ 0; VS.2 ~~ VS.2; pVS.2 ~~ 0*pVS.2
pVS.3 =~ 1*VS.3; VS.3 ~ 0; VS.3 ~~ VS.3; pVS.3 ~~ 0*pVS.3
pVS.4 =~ 1*VS.4; VS.4 ~ 0; VS.4 ~~ VS.4; pVS.4 ~~ 0*pVS.4
pVS.5 =~ 1*VS.5; VS.5 ~ 0; VS.5 ~~ VS.5; pVS.5 ~~ 0*pVS.5
# Regressions Between Adjacent Observations
pVS.2 ~ 1*pVS.1
pVS.3 ~ 1*pVS.2
pVS.4 ~ 1*pVS.3
pVS.5 ~ 1*pVS.4
# Define Change Latent Variables
dVS.21 =~ 1*pVS.2; dVS.21 ~~ 0*dVS.21
dVS.32 =~ 1*pVS.3; dVS.32 ~~ 0*dVS.32
dVS.43 =~ 1*pVS.4; dVS.43 ~~ 0*dVS.43
dVS.54 =~ 1*pVS.5; dVS.54 ~~ 0*dVS.54
# Define Proportional Regressions
dVS.21 ~ beta*pVS.1
dVS.32 ~ beta*pVS.2
dVS.43 ~ beta*pVS.3
dVS.54 ~ beta*pVS.4
# Define Intercept and Slope
int =~ 1*pVS.1
slp =~ 1*dVS.21 + 1*dVS.32 + 1*dVS.43 + 1*dVS.54
int ~ 1
slp ~ 1
int ~~ int + slp
slp ~~ slp
'
LCSpt.fit = sem(LCSpt, data=ABCD, missing='ML')
summary(LCSpt.fit)## lavaan 0.6-7 ended normally after 72 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 14
## Number of equality constraints 3
##
## Number of observations 964
## Number of missing patterns 8
##
## Model Test User Model:
##
## Test statistic 26.912
## Degrees of freedom 9
## P-value (Chi-square) 0.001
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## pVS.1 =~
## VS.1 1.000
## pVS.2 =~
## VS.2 1.000
## pVS.3 =~
## VS.3 1.000
## pVS.4 =~
## VS.4 1.000
## pVS.5 =~
## VS.5 1.000
## dVS.21 =~
## pVS.2 1.000
## dVS.32 =~
## pVS.3 1.000
## dVS.43 =~
## pVS.4 1.000
## dVS.54 =~
## pVS.5 1.000
## int =~
## pVS.1 1.000
## slp =~
## dVS.21 1.000
## dVS.32 1.000
## dVS.43 1.000
## dVS.54 1.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## pVS.2 ~
## pVS.1 1.000
## pVS.3 ~
## pVS.2 1.000
## pVS.4 ~
## pVS.3 1.000
## pVS.5 ~
## pVS.4 1.000
## dVS.21 ~
## pVS.1 (beta) 0.298 0.105 2.852 0.004
## dVS.32 ~
## pVS.2 (beta) 0.298 0.105 2.852 0.004
## dVS.43 ~
## pVS.3 (beta) 0.298 0.105 2.852 0.004
## dVS.54 ~
## pVS.4 (beta) 0.298 0.105 2.852 0.004
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## int ~~
## slp -0.278 0.102 -2.716 0.007
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## int 6.256 0.045 139.992 0.000
## slp -1.739 0.682 -2.551 0.011
## .pVS.1 0.000
## .pVS.2 0.000
## .pVS.3 0.000
## .pVS.4 0.000
## .pVS.5 0.000
## .dVS.21 0.000
## .dVS.32 0.000
## .dVS.43 0.000
## .dVS.54 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.316 0.084 15.693 0.000
## .pVS.1 0.000
## .VS.2 1.348 0.077 17.558 0.000
## .pVS.2 0.000
## .VS.3 1.104 0.061 18.056 0.000
## .pVS.3 0.000
## .VS.4 0.999 0.062 16.185 0.000
## .pVS.4 0.000
## .VS.5 0.163 0.091 1.788 0.074
## .pVS.5 0.000
## .dVS.21 0.000
## .dVS.32 0.000
## .dVS.43 0.000
## .dVS.54 0.000
## int 0.949 0.074 12.796 0.000
## slp 0.119 0.049 2.452 0.0145.10 Bivariate Latent Change Score Model
# Bivariate Within-Construct Proportional LCS Trajectory Model
bLCSt = '# Define Phantom Variables
pVS.1 =~ 1*VS.1; VS.1 ~ 0; VS.1 ~~ VS.1; pVS.1 ~~ 0*pVS.1
pVS.2 =~ 1*VS.2; VS.2 ~ 0; VS.2 ~~ VS.2; pVS.2 ~~ 0*pVS.2
pVS.3 =~ 1*VS.3; VS.3 ~ 0; VS.3 ~~ VS.3; pVS.3 ~~ 0*pVS.3
pVS.4 =~ 1*VS.4; VS.4 ~ 0; VS.4 ~~ VS.4; pVS.4 ~~ 0*pVS.4
pVS.5 =~ 1*VS.5; VS.5 ~ 0; VS.5 ~~ VS.5; pVS.5 ~~ 0*pVS.5
pEXT.1 =~ 1*EXT.1; EXT.1 ~ 0; EXT.1 ~~ EXT.1; pEXT.1 ~~ 0*pEXT.1
pEXT.2 =~ 1*EXT.2; EXT.2 ~ 0; EXT.2 ~~ EXT.2; pEXT.2 ~~ 0*pEXT.2
pEXT.3 =~ 1*EXT.3; EXT.3 ~ 0; EXT.3 ~~ EXT.3; pEXT.3 ~~ 0*pEXT.3
pEXT.4 =~ 1*EXT.4; EXT.4 ~ 0; EXT.4 ~~ EXT.4; pEXT.4 ~~ 0*pEXT.4
pEXT.5 =~ 1*EXT.5; EXT.5 ~ 0; EXT.5 ~~ EXT.5; pEXT.5 ~~ 0*pEXT.5
# Residual Cross-Construct Covariances
VS.1 ~~ EXT.1
VS.2 ~~ EXT.2
VS.3 ~~ EXT.3
VS.4 ~~ EXT.4
VS.5 ~~ EXT.5
# Regressions Between Adjacent Observations
pVS.2 ~ 1*pVS.1
pVS.3 ~ 1*pVS.2
pVS.4 ~ 1*pVS.3
pVS.5 ~ 1*pVS.4
pEXT.2 ~ 1*pEXT.1
pEXT.3 ~ 1*pEXT.2
pEXT.4 ~ 1*pEXT.3
pEXT.5 ~ 1*pEXT.4
# Define Change Latent Variables
dVS.21 =~ 1*pVS.2; dVS.21 ~~ 0*dVS.21
dVS.32 =~ 1*pVS.3; dVS.32 ~~ 0*dVS.32
dVS.43 =~ 1*pVS.4; dVS.43 ~~ 0*dVS.43
dVS.54 =~ 1*pVS.5; dVS.54 ~~ 0*dVS.54
dEXT.21 =~ 1*pEXT.2; dEXT.21 ~~ 0*dEXT.21
dEXT.32 =~ 1*pEXT.3; dEXT.32 ~~ 0*dEXT.32
dEXT.43 =~ 1*pEXT.4; dEXT.43 ~~ 0*dEXT.43
dEXT.54 =~ 1*pEXT.5; dEXT.54 ~~ 0*dEXT.54
# Define Within-Construct Proportional Regressions
dVS.21 ~ beta.V*pVS.1
dVS.32 ~ beta.V*pVS.2
dVS.43 ~ beta.V*pVS.3
dVS.54 ~ beta.V*pVS.4
dEXT.21 ~ beta.E*pEXT.1
dEXT.32 ~ beta.E*pEXT.2
dEXT.43 ~ beta.E*pEXT.3
dEXT.54 ~ beta.E*pEXT.4
# Define Intercept and Slope
int.V =~ 1*pVS.1
slp.V =~ 1*dVS.21 + 1*dVS.32 + 1*dVS.43 + 1*dVS.54
int.V ~ 1
slp.V ~ 1
int.V ~~ int.V + slp.V + int.E + slp.E
slp.V ~~ slp.V + int.E + slp.E
int.E =~ 1*pEXT.1
slp.E =~ 1*dEXT.21 + 1*dEXT.32 + 1*dEXT.43 + 1*dEXT.54
int.E ~ 1
slp.E ~ 1
int.E ~~ int.E + slp.E
slp.E ~~ slp.E
'
bLCSt.fit = sem(bLCSt, data=ABCD, missing='ML')
summary(bLCSt.fit)## lavaan 0.6-7 ended normally after 121 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 37
## Number of equality constraints 6
##
## Number of observations 964
## Number of missing patterns 47
##
## Model Test User Model:
##
## Test statistic 74.560
## Degrees of freedom 34
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## pVS.1 =~
## VS.1 1.000
## pVS.2 =~
## VS.2 1.000
## pVS.3 =~
## VS.3 1.000
## pVS.4 =~
## VS.4 1.000
## pVS.5 =~
## VS.5 1.000
## pEXT.1 =~
## EXT.1 1.000
## pEXT.2 =~
## EXT.2 1.000
## pEXT.3 =~
## EXT.3 1.000
## pEXT.4 =~
## EXT.4 1.000
## pEXT.5 =~
## EXT.5 1.000
## dVS.21 =~
## pVS.2 1.000
## dVS.32 =~
## pVS.3 1.000
## dVS.43 =~
## pVS.4 1.000
## dVS.54 =~
## pVS.5 1.000
## dEXT.21 =~
## pEXT.2 1.000
## dEXT.32 =~
## pEXT.3 1.000
## dEXT.43 =~
## pEXT.4 1.000
## dEXT.54 =~
## pEXT.5 1.000
## int.V =~
## pVS.1 1.000
## slp.V =~
## dVS.21 1.000
## dVS.32 1.000
## dVS.43 1.000
## dVS.54 1.000
## int.E =~
## pEXT.1 1.000
## slp.E =~
## dEXT.21 1.000
## dEXT.32 1.000
## dEXT.43 1.000
## dEXT.54 1.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## pVS.2 ~
## pVS.1 1.000
## pVS.3 ~
## pVS.2 1.000
## pVS.4 ~
## pVS.3 1.000
## pVS.5 ~
## pVS.4 1.000
## pEXT.2 ~
## pEXT.1 1.000
## pEXT.3 ~
## pEXT.2 1.000
## pEXT.4 ~
## pEXT.3 1.000
## pEXT.5 ~
## pEXT.4 1.000
## dVS.21 ~
## pVS.1 (bt.V) 0.348 0.108 3.211 0.001
## dVS.32 ~
## pVS.2 (bt.V) 0.348 0.108 3.211 0.001
## dVS.43 ~
## pVS.3 (bt.V) 0.348 0.108 3.211 0.001
## dVS.54 ~
## pVS.4 (bt.V) 0.348 0.108 3.211 0.001
## dEXT.21 ~
## pEXT.1 (bt.E) 0.365 0.158 2.319 0.020
## dEXT.32 ~
## pEXT.2 (bt.E) 0.365 0.158 2.319 0.020
## dEXT.43 ~
## pEXT.3 (bt.E) 0.365 0.158 2.319 0.020
## dEXT.54 ~
## pEXT.4 (bt.E) 0.365 0.158 2.319 0.020
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 ~~
## .EXT.1 -0.024 0.030 -0.797 0.425
## .VS.2 ~~
## .EXT.2 0.091 0.026 3.563 0.000
## .VS.3 ~~
## .EXT.3 0.120 0.027 4.505 0.000
## .VS.4 ~~
## .EXT.4 0.094 0.021 4.547 0.000
## .VS.5 ~~
## .EXT.5 -0.021 0.039 -0.532 0.595
## int.V ~~
## slp.V -0.319 0.107 -2.977 0.003
## int.E 0.044 0.032 1.372 0.170
## slp.E 0.005 0.021 0.252 0.801
## slp.V ~~
## int.E 0.019 0.019 1.005 0.315
## slp.E -0.018 0.008 -2.303 0.021
## int.E ~~
## slp.E -0.156 0.063 -2.488 0.013
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## .EXT.1 0.000
## .EXT.2 0.000
## .EXT.3 0.000
## .EXT.4 0.000
## .EXT.5 0.000
## int.V 6.267 0.044 141.086 0.000
## slp.V -2.063 0.705 -2.925 0.003
## int.E 2.342 0.026 91.672 0.000
## slp.E -0.810 0.380 -2.131 0.033
## .pVS.1 0.000
## .pVS.2 0.000
## .pVS.3 0.000
## .pVS.4 0.000
## .pVS.5 0.000
## .pEXT.1 0.000
## .pEXT.2 0.000
## .pEXT.3 0.000
## .pEXT.4 0.000
## .pEXT.5 0.000
## .dVS.21 0.000
## .dVS.32 0.000
## .dVS.43 0.000
## .dVS.54 0.000
## .dEXT.21 0.000
## .dEXT.32 0.000
## .dEXT.43 0.000
## .dEXT.54 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.338 0.084 15.967 0.000
## .pVS.1 0.000
## .VS.2 1.339 0.076 17.546 0.000
## .pVS.2 0.000
## .VS.3 1.110 0.061 18.093 0.000
## .pVS.3 0.000
## .VS.4 1.018 0.062 16.406 0.000
## .pVS.4 0.000
## .VS.5 0.118 0.091 1.293 0.196
## .pVS.5 0.000
## .EXT.1 0.307 0.023 13.433 0.000
## .pEXT.1 0.000
## .EXT.2 0.254 0.016 15.388 0.000
## .pEXT.2 0.000
## .EXT.3 0.372 0.021 17.657 0.000
## .pEXT.3 0.000
## .EXT.4 0.151 0.015 9.718 0.000
## .pEXT.4 0.000
## .EXT.5 0.819 0.056 14.520 0.000
## .pEXT.5 0.000
## .dVS.21 0.000
## .dVS.32 0.000
## .dVS.43 0.000
## .dVS.54 0.000
## .dEXT.21 0.000
## .dEXT.32 0.000
## .dEXT.43 0.000
## .dEXT.54 0.000
## int.V 0.942 0.073 12.904 0.000
## slp.V 0.141 0.061 2.308 0.021
## int.E 0.415 0.028 14.643 0.000
## slp.E 0.068 0.043 1.565 0.118# Bivariate Dual Change LCS Trajectory Model
bLCSpt = '# Define Phantom Variables
pVS.1 =~ 1*VS.1; VS.1 ~ 0; VS.1 ~~ VS.1; pVS.1 ~~ 0*pVS.1
pVS.2 =~ 1*VS.2; VS.2 ~ 0; VS.2 ~~ VS.2; pVS.2 ~~ 0*pVS.2
pVS.3 =~ 1*VS.3; VS.3 ~ 0; VS.3 ~~ VS.3; pVS.3 ~~ 0*pVS.3
pVS.4 =~ 1*VS.4; VS.4 ~ 0; VS.4 ~~ VS.4; pVS.4 ~~ 0*pVS.4
pVS.5 =~ 1*VS.5; VS.5 ~ 0; VS.5 ~~ VS.5; pVS.5 ~~ 0*pVS.5
pEXT.1 =~ 1*EXT.1; EXT.1 ~ 0; EXT.1 ~~ EXT.1; pEXT.1 ~~ 0*pEXT.1
pEXT.2 =~ 1*EXT.2; EXT.2 ~ 0; EXT.2 ~~ EXT.2; pEXT.2 ~~ 0*pEXT.2
pEXT.3 =~ 1*EXT.3; EXT.3 ~ 0; EXT.3 ~~ EXT.3; pEXT.3 ~~ 0*pEXT.3
pEXT.4 =~ 1*EXT.4; EXT.4 ~ 0; EXT.4 ~~ EXT.4; pEXT.4 ~~ 0*pEXT.4
pEXT.5 =~ 1*EXT.5; EXT.5 ~ 0; EXT.5 ~~ EXT.5; pEXT.5 ~~ 0*pEXT.5
# Residual Cross-Construct Covariances
VS.1 ~~ EXT.1
VS.2 ~~ EXT.2
VS.3 ~~ EXT.3
VS.4 ~~ EXT.4
VS.5 ~~ EXT.5
# Regressions Between Adjacent Observations
pVS.2 ~ 1*pVS.1
pVS.3 ~ 1*pVS.2
pVS.4 ~ 1*pVS.3
pVS.5 ~ 1*pVS.4
pEXT.2 ~ 1*pEXT.1
pEXT.3 ~ 1*pEXT.2
pEXT.4 ~ 1*pEXT.3
pEXT.5 ~ 1*pEXT.4
# Define Change Latent Variables
dVS.21 =~ 1*pVS.2; dVS.21 ~~ 0*dVS.21
dVS.32 =~ 1*pVS.3; dVS.32 ~~ 0*dVS.32
dVS.43 =~ 1*pVS.4; dVS.43 ~~ 0*dVS.43
dVS.54 =~ 1*pVS.5; dVS.54 ~~ 0*dVS.54
dEXT.21 =~ 1*pEXT.2; dEXT.21 ~~ 0*dEXT.21
dEXT.32 =~ 1*pEXT.3; dEXT.32 ~~ 0*dEXT.32
dEXT.43 =~ 1*pEXT.4; dEXT.43 ~~ 0*dEXT.43
dEXT.54 =~ 1*pEXT.5; dEXT.54 ~~ 0*dEXT.54
# Define Within- and Between Construct Proportional Regressions
dVS.21 ~ beta.V*pVS.1 + beta.VE*pEXT.1
dVS.32 ~ beta.V*pVS.2 + beta.VE*pEXT.2
dVS.43 ~ beta.V*pVS.3 + beta.VE*pEXT.3
dVS.54 ~ beta.V*pVS.4 + beta.VE*pEXT.4
dEXT.21 ~ beta.E*pEXT.1 + beta.EV*pVS.1
dEXT.32 ~ beta.E*pEXT.2 + beta.EV*pVS.2
dEXT.43 ~ beta.E*pEXT.3 + beta.EV*pVS.3
dEXT.54 ~ beta.E*pEXT.4 + beta.EV*pVS.4
# Define Intercept and Slope
int.V =~ 1*pVS.1
slp.V =~ 1*dVS.21 + 1*dVS.32 + 1*dVS.43 + 1*dVS.54
int.V ~ 1
slp.V ~ 1
int.V ~~ int.V + slp.V + int.E + slp.E
slp.V ~~ slp.V + int.E + slp.E
int.E =~ 1*pEXT.1
slp.E =~ 1*dEXT.21 + 1*dEXT.32 + 1*dEXT.43 + 1*dEXT.54
int.E ~ 1
slp.E ~ 1
int.E ~~ int.E + slp.E
slp.E ~~ slp.E
'
bLCSpt.fit = sem(bLCSpt, data=ABCD, missing='ML')
summary(bLCSpt.fit)## lavaan 0.6-7 ended normally after 177 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 45
## Number of equality constraints 12
##
## Number of observations 964
## Number of missing patterns 47
##
## Model Test User Model:
##
## Test statistic 74.006
## Degrees of freedom 32
## P-value (Chi-square) 0.000
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Observed
## Observed information based on Hessian
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## pVS.1 =~
## VS.1 1.000
## pVS.2 =~
## VS.2 1.000
## pVS.3 =~
## VS.3 1.000
## pVS.4 =~
## VS.4 1.000
## pVS.5 =~
## VS.5 1.000
## pEXT.1 =~
## EXT.1 1.000
## pEXT.2 =~
## EXT.2 1.000
## pEXT.3 =~
## EXT.3 1.000
## pEXT.4 =~
## EXT.4 1.000
## pEXT.5 =~
## EXT.5 1.000
## dVS.21 =~
## pVS.2 1.000
## dVS.32 =~
## pVS.3 1.000
## dVS.43 =~
## pVS.4 1.000
## dVS.54 =~
## pVS.5 1.000
## dEXT.21 =~
## pEXT.2 1.000
## dEXT.32 =~
## pEXT.3 1.000
## dEXT.43 =~
## pEXT.4 1.000
## dEXT.54 =~
## pEXT.5 1.000
## int.V =~
## pVS.1 1.000
## slp.V =~
## dVS.21 1.000
## dVS.32 1.000
## dVS.43 1.000
## dVS.54 1.000
## int.E =~
## pEXT.1 1.000
## slp.E =~
## dEXT.21 1.000
## dEXT.32 1.000
## dEXT.43 1.000
## dEXT.54 1.000
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## pVS.2 ~
## pVS.1 1.000
## pVS.3 ~
## pVS.2 1.000
## pVS.4 ~
## pVS.3 1.000
## pVS.5 ~
## pVS.4 1.000
## pEXT.2 ~
## pEXT.1 1.000
## pEXT.3 ~
## pEXT.2 1.000
## pEXT.4 ~
## pEXT.3 1.000
## pEXT.5 ~
## pEXT.4 1.000
## dVS.21 ~
## pVS.1 (bt.V) 0.322 0.120 2.682 0.007
## pEXT.1 (b.VE) 0.118 0.253 0.465 0.642
## dVS.32 ~
## pVS.2 (bt.V) 0.322 0.120 2.682 0.007
## pEXT.2 (b.VE) 0.118 0.253 0.465 0.642
## dVS.43 ~
## pVS.3 (bt.V) 0.322 0.120 2.682 0.007
## pEXT.3 (b.VE) 0.118 0.253 0.465 0.642
## dVS.54 ~
## pVS.4 (bt.V) 0.322 0.120 2.682 0.007
## pEXT.4 (b.VE) 0.118 0.253 0.465 0.642
## dEXT.21 ~
## pEXT.1 (bt.E) 0.324 0.166 1.955 0.051
## pVS.1 (b.EV) 0.047 0.067 0.708 0.479
## dEXT.32 ~
## pEXT.2 (bt.E) 0.324 0.166 1.955 0.051
## pVS.2 (b.EV) 0.047 0.067 0.708 0.479
## dEXT.43 ~
## pEXT.3 (bt.E) 0.324 0.166 1.955 0.051
## pVS.3 (b.EV) 0.047 0.067 0.708 0.479
## dEXT.54 ~
## pEXT.4 (bt.E) 0.324 0.166 1.955 0.051
## pVS.4 (b.EV) 0.047 0.067 0.708 0.479
##
## Covariances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 ~~
## .EXT.1 -0.016 0.033 -0.479 0.632
## .VS.2 ~~
## .EXT.2 0.089 0.026 3.495 0.000
## .VS.3 ~~
## .EXT.3 0.119 0.027 4.465 0.000
## .VS.4 ~~
## .EXT.4 0.099 0.023 4.353 0.000
## .VS.5 ~~
## .EXT.5 -0.045 0.049 -0.911 0.362
## int.V ~~
## slp.V -0.309 0.108 -2.866 0.004
## int.E 0.041 0.033 1.228 0.219
## slp.E -0.034 0.061 -0.555 0.579
## slp.V ~~
## int.E -0.024 0.098 -0.247 0.805
## slp.E 0.008 0.037 0.206 0.837
## int.E ~~
## slp.E -0.145 0.063 -2.290 0.022
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 0.000
## .VS.2 0.000
## .VS.3 0.000
## .VS.4 0.000
## .VS.5 0.000
## .EXT.1 0.000
## .EXT.2 0.000
## .EXT.3 0.000
## .EXT.4 0.000
## .EXT.5 0.000
## int.V 6.270 0.045 139.286 0.000
## slp.V -2.184 0.743 -2.941 0.003
## int.E 2.345 0.026 89.978 0.000
## slp.E -1.018 0.475 -2.145 0.032
## .pVS.1 0.000
## .pVS.2 0.000
## .pVS.3 0.000
## .pVS.4 0.000
## .pVS.5 0.000
## .pEXT.1 0.000
## .pEXT.2 0.000
## .pEXT.3 0.000
## .pEXT.4 0.000
## .pEXT.5 0.000
## .dVS.21 0.000
## .dVS.32 0.000
## .dVS.43 0.000
## .dVS.54 0.000
## .dEXT.21 0.000
## .dEXT.32 0.000
## .dEXT.43 0.000
## .dEXT.54 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .VS.1 1.327 0.087 15.280 0.000
## .pVS.1 0.000
## .VS.2 1.341 0.076 17.545 0.000
## .pVS.2 0.000
## .VS.3 1.110 0.061 18.118 0.000
## .pVS.3 0.000
## .VS.4 1.014 0.063 16.025 0.000
## .pVS.4 0.000
## .VS.5 0.134 0.098 1.377 0.169
## .pVS.5 0.000
## .EXT.1 0.304 0.024 12.712 0.000
## .pEXT.1 0.000
## .EXT.2 0.254 0.017 15.399 0.000
## .pEXT.2 0.000
## .EXT.3 0.372 0.021 17.670 0.000
## .pEXT.3 0.000
## .EXT.4 0.149 0.016 9.527 0.000
## .pEXT.4 0.000
## .EXT.5 0.829 0.057 14.604 0.000
## .pEXT.5 0.000
## .dVS.21 0.000
## .dVS.32 0.000
## .dVS.43 0.000
## .dVS.54 0.000
## .dEXT.21 0.000
## .dEXT.32 0.000
## .dEXT.43 0.000
## .dEXT.54 0.000
## int.V 0.951 0.076 12.514 0.000
## slp.V 0.135 0.058 2.339 0.019
## int.E 0.418 0.030 14.024 0.000
## slp.E 0.061 0.039 1.545 0.122